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					    REGIONAL
  MATHEMATICS
CURRICULUM GUIDE


  DEVELOPED BY AND FOR:


THE REGIONAL DISTRICTS OF
FRANKFORD, LAFAYETTE, AND
 SUSSEX-WANTAGE SCHOOLS



          2008
             TABLE OF CONTENTS

                                                    Page
Credits………………………………………………………………3

Philosophy…………………………………………………………..4

Implementation……………………………………………………...5

Assessment……………………………………………….…………6

Standards……………………………………………………………7

Instructional Units (Pre-School – Grade 8)………………………..12

Appendix
    NJ CCCS Mathematics Revisions…………………………..187
    Standards 4.1 – 4.5: Vertical Alignment…………………...190
    Internet Websites……………………………………………221




                             2
                           CREDITS


Grateful recognition is made to the following individuals
     for their level of expertise and dedicated work.

                         MEMBERS:

Kim Branham – Lafayette                 June Yucius – Frankford
Tara MacGlashan – Lafayette             Carole Flynn – Frankford
Allyn Perry – Lafayette                 Cathy Gardner – Frankford
Linda Jensen – Lafayette                Jane Fialcowitz – Frankford
Nicole Worthington – Lafayette          Judy Gray – Frankford
Anne Lajeunesse – Lafayette             Linda Suchana – Frankford
Carol Wilson – Sussex-Wantage           Jodi Kuzmiak – Frankford
Joan Casserly – Sussex-Wantage          Darcy Harris – Frankford
Cathryn Weiss – Sussex-Wantage          Sara Beattie – Frankford
Maureen Vatalaro – Sussex-Wantage




     Appreciation to the following members for their
               organization and guidance:

                Genene Pagliaro – Frankford / Lafayette
                   Susan Petrick – Sussex-Wantage



                                    3
                              PHILOSOPHY


All students should be afforded the opportunity to achieve mathematics
proficiency through an integration of understanding, comprehending,
applying, reasoning, and analyzing. The math opportunity for students must
be rich and complex and connected to real life experiences. Through a
specific math language, students will be able to communicate
comprehension in both oral and written form.

We must enable all of our children to acquire math skills, understanding, and
attitudes that they will need to be successful in their careers and daily lives.
We believe all students can learn math and all students need to learn math.

The local community, educators, parents, and students shall work together to
make the revision of the NJ Mathematics Standards a reality.




                                       4
                         IMPLEMENTATION



The New Jersey Core Curriculum Content Standards provided the basis for
the development of this curriculum guide. In order for implementation of
this curriculum to occur, the following need to be in place:

1. Adequate instructional time;

2. Infusion of mathematics concepts in all content areas;

3. Effective instructional materials, including all necessary supplemental
and technological support and resources;

4. Consistent and sustained Professional Development for all teachers of
mathematics.




                                      5
                                     ASSESSMENT


THE ASSESSMENT PROCESS: Assessment is a way of providing feedback to the
various stakeholders in the educational system, and of communicating the outcomes to all
concerned. The data provide feedback to:

                students on how well they are meeting expectations
                teachers with how well students are learning
                districts on the effectiveness of their programs
                policy makers on how well policies are working.



CLASSROOM ASSESSMENT: Classroom teachers should utilize a variety of assessment
tools designed to provide information on student comprehension and progress toward learning
objectives. Assessment should be based upon, but not limited to, the following:

              1.   Open-ended problems
              2.   Teacher interviews
              3.   Portfolios
              4.   Mathematical journals
              5.   Formative and summative assessments
              6.   Completion of assignments, both in and out of the classroom
              7.   Oral contribution in class
              8.   Rubrics




                                              6
                       Mathematics Standards
                              2002

The revised standards adopted in 2002 are more specific and clearer than the
1996 standards. The 16 standards of 1996 have been organized into 5
standards that correspond to the content clusters of the statewide
assessments. There are student expectations at each grade level beginning
with kindergarten. Standards 1 – 4 define the content that students should
know and be able to do.

4.1.   Number and Numerical Operations
       A. Number Sense
       B. Numerical Operations
       C. Estimation

4.2.   Geometry and Measurement
       A. Geometric Properties
       B. Transforming Shapes
       C. Coordinate Geometry
       D. Units of Measurement
       E. Measuring Geometric Objects

4.3.   Patterns and Algebra
       A. Patterns and Relationships
       B. Functions
       C. Modeling
       D. Procedures

4.4.   Data Analysis, Probability, and Discrete Mathematics
       A. Data Analysis (Statistics)
       B. Probability
       C. Discrete Mathematics—Systematic Listing and Counting
       D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms

The fifth standard defines the mathematical processes that all students at
each grade level should use when acquiring and applying content knowledge
and skills of the first four standards.




                                       7
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                      8
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                      9
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             10
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       11
                                                MATHEMATICS
                                                 PRE-SCHOOL


        EXPECTATION                           STUDENT OUTCOME                             TEACHER’S NOTES AND
                                                                                            SUPPLEMENTARY
                                                                                              RESOURCES
EXPECTATION 1:                1.1 Demonstrates understanding of one-to-one
Children demonstrate an       correspondence (e.g., places one placemat at each place,
understanding of number and   gives each child one cookie, places one animal in each
numerical operations.         truck, hands out manipulatives to be shared with a friend
                              saying "One for you, one for me.").

                              1.2 Spontaneously counts for own purposes (e.g.,
                              counting blocks or cars, counting beads while stringing
                              them, handing out napkins).

                              1.3 Learns to say the counting numbers.

                              1.4 Discriminates numbers from other symbols in the
                              environment (e.g., street signs, license plates, room
                              number, clock, etc.).

                              1.5 Recognizes and names some written numerals.

                              1.6 Compares numbers in different contexts (e.g., using
                              words such as more and less).

                              1.7 Uses estimation as a method for approximating an
                              appropriate amount (e.g., at snack time, deciding how
                              many napkins to take from a large pile for the group,
                              determining number of blocks to use when building
                              structures).


                                                       12
                                     1.8 Adds two groups of concrete objects by counting the
                                     total (e.g., three blue pegs, three yellow pegs, six pegs
                                     altogether).
                                     1.9 Subtracts one group of concrete objects from another
                                     by taking some away and then counting the remainder
                                     (e.g., "I have four carrot sticks. I'm eating one! Now I
                                     have 3!").
         EXPECTATION 2:              2.1 Identifies basic shapes in the environment (e.g., circle,
Children develop knowledge of        square, triangle, cube, sphere).
spatial concepts, e.g., shapes and
measurement.                         2.2 Uses standard and nonstandard measurement units
                                     (e.g., measuring body length with unifix cubes, using a
                                     tape measure to gauge height of block construction,
                                     counting the number of cups it takes to fill a bucket with
                                     water).

                                     2.3 Uses vocabulary to describe distances (e.g., "It was a
                                     really long walk to the playground.").

                                     2.4 Uses vocabulary to describe directional concept (e.g.,
                                     "Watch me climb up the ladder and slide down.").

                                     2.5 Uses positional words in a functional way (e.g., "I put
                                     the red block on top of the cabinet.").

                                     2.6 Makes three-dimensional constructions and models
                                     (e.g., sculptures that have height, depth and width).

                                     2.7 Makes connections between two dimensional and
                                     three dimensional forms (e.g., circle-sphere, square-cube,
                                     triangle-pyramid).

EXPECTATION 3:                       3.1 Sorts objects into groups (e.g., separate basket of
Children understand patterns,        collected items into piles of pinecones, acorns and twigs).
relationships and classification.

                                                               13
                                   3.2 Classifies objects by sorting them into subgroups by
                                   one or more attributes (e.g., sorting counting bears by
                                   color into trays, separating a mixture of beans by
                                   individual size and shape).

                                   3.3 Describes an object by characteristics it does or does
                                   not possess (e.g., "This button doesn't have holes.").

                                   3.4 Seriates objects according to various properties
                                   including size, number, length, heaviness, texture (rough
                                   to smooth) or loudness.

                                   3.5 Identifies patterns in the environment (e.g., "Look at
                                   the rug. It has a circle, then a number, then a letter...").

                                   3.6 Represents patterns in a variety of ways (e.g.,
                                   stringing beads red/green/red/green/red/green, arranging
                                   buttons big/bigger/biggest, or singing songs that follow a
                                   simple pattern).
EXPECTATION 4:                     4.1 Starts and stops on a signal (e.g., freezing in position
Children develop knowledge of      when the music stops).
sequence and temporal awareness.
                                   4.2 Describes the sequence of the daily routine and
                                   demonstrates understanding of basic temporal relations
                                   (e.g., "We will go outside after snack time.").

                                   4.3 Arranges pictures of events in temporal order (e.g.,
                                   first, a photo of the child eating breakfast; second, a photo
                                   of the child getting on the bus; third, a photo of the child
                                   in the classroom).




                                                             14
EXPECTATION 5:                             5.1 Uses mathematical terms when conversing with
Children will use mathematical             others (e.g., "Which car is faster?" "My building is
knowledge to represent, communicate        taller than yours." "I have more sand in my
and solve problems in their environment.   bucket.").

                                           5.2 Uses emergent mathematical knowledge as a
                                           problem-solving tool (e.g., Maritza notices that Juan
                                           has more carrot sticks than she does. She says, "May
                                           I have some of yours? Then we will have the same
                                           amount." Jorge decides to fill his bucket by using
                                           small cups of water when he realizes that he cannot
                                           fit the bucket under the faucet).

                                           5.3 Describes how he/she solved mathematical
                                           problems in his/her own way.




                                                              15
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                      16
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                      17
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             18
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       19
                                                         MATHEMATICS
                                                         KINDERGARTEN

                                                                                                                 TEACHER’S NOTES &
         STANDARD                     STUDENT OUTCOME                       SUGGESTED ACTIVITIES                  SUPPLEMENTARY
                                                                              NJ FRAMEWORKS 1996                    RESOURCES
4.1 All students will develop      4.1.A. Number Sense                  Pgs. 176-177, Overview
    number sense and will
    perform standard numerical     4.1.A.1 – Use real-life              Pg. 178, No. 1: Use real-life
    operations and estimations     experiences, physical                experiences, physical materials, and
    on all types of numbers in a   materials, and technology to         technology to construct meanings for
    variety of ways.               construct meanings for               whole numbers, commonly used
4.1.A. Number Sense                numbers.                             fractions, and decimals.
4.1.B. Numerical Operations            • Whole numbers to 20
4.1.C. Estimation                      • Ordinals to 10
                                       • Proper fractions
                                          (denominators of 2;
                                          concept of 1/2)

                                   4.1.A.2 – Demonstrate an             Pg. 179, No. 4: Develop a sense of the
                                   understanding of whole               magnitudes of whole numbers and
                                   number place value concepts          commonly used fractions.
                                   (tens and ones—units).

                                   4.1.A.3 – Understand that       Pg. 180, No. 5: Understand the
                                   numbers have a variety of uses. various uses of numbers including
                                                                   counting, measuring, labeling, and
                                                                   indicating location.

                                   4.1.A.4 – Count and perform
                                   simple computations with
                                   coins.
                                       • Amounts up to $1.00
                                          (using cents notation)
                                   (Recognition of four basic

                                                                   20
coins)

4.1.A.5 – Compare and order        Pg. 181, No. 8: Compare and order
whole numbers.                     whole numbers, commonly used
                                   fractions, and decimals.


4.1.B. Numerical Operations        Pgs. 253-255, Overview

4.1.B.1 – Develop the meanings     Pg. 256, No. 1: Develop meaning for
of addition and subtraction by     the four basic arithmetic operations by
concretely modeling and            modeling and discussing a variety of
discussing a large variety of      problems.
problems.                          Pg. 257, No. 2: Develop proficiency
    • Joining, separating, and     with and memorize basic number facts
       comparing                   using a variety of fact strategies (such
                                   as “counting on” and “doubles”).
                                   Pg. 257, No. 3: Construct, use and
                                   explain procedures for performing
                                   whole number calculations in the
                                   various methods of computation.
                                   Pg. 258, No. 5: Use a variety of
                                   mental computation and estimation
                                   techniques.
                                   Pg. 259, No. 6: Select and use
                                   appropriate computational methods
                                   from mental math, estimation, paper-
                                   and-pencil, and calculator methods,
                                   and check, the reasonableness of
                                   results.
                                   Pg. 259, No. 7: Understand and use
                                   relationships among operations and
                                   properties of operations.



                              21
                                    4.1.C. Estimation                   Pgs. 300-310, Overview

                                    4.1.C.1 – Judge without             Pg. 312, No. 1: Judge without
                                    counting whether a set of           counting whether a set of objects has
                                    objects has less than, more         less than, more than, or the same
                                    than, or the same number of         number of objects as a reference set.
                                    objects as a reference set.

                                    4.1.C.3 – Explore a variety of      Pg. 313, No. 4: Explore, construct,
                                    strategies for estimating both      and use a variety of estimation
                                    quantities (e.g., the number of     strategies.
                                    marbles in a jar) and results of    Pg. 258, No. 5: Use a variety of
                                    computation.                        mental computation and estimation
                                                                        techniques.
                                                                        Pg. 259, No. 6: Select and use
                                                                        appropriate computational methods
                                                                        from mental math, estimation, paper-
                                                                        and-pencil, and calculator methods,
                                                                        and check the reasonableness of
                                                                        results.
                                                                        Pg. 288, No. 6: Understand and
                                                                        incorporate estimation and repeated
                                                                        measures in measurement activities.


4.2 All students will develop       4.2.A. Geometric Properties         Pgs. 213-214, Overview
    spatial sense and the ability
    to use geometric properties,    4.2.A.1 – Identify and describe     Pg. 215, No. 1: Explore spatial
    relationships, and              spatial relationships among         relationships such as the direction,
    measurement to model,           objects in space and their          orientation, and perspectives of objects
    describe and analyze            relative shapes and sizes.          in space, their relative shapes and
    phenomena.                          • Inside/outside,               sizes, and the relations between objects
4.2.A. Geometric Properties                 left/right, above/below,    and their shadows or projections.
4.2.B. Transforming Shapes                  between
4.2.D. Units of Measurement             • Smaller/larger/same

                                                                   22
       size, wider/narrower,
       longer/shorter
   •   Congruence (i.e. same
       size and shape)

4.2.A.2 – Use concrete objects,       Pg. 215, No. 3: Explore properties of
drawings, and computer                three- and two-dimensional shapes
graphics to identify, classify,       using concrete objects, drawings, and
and describe standard three-          computer graphics.
dimensional and two-                  Pg. 216, No. 4: Use properties of
dimensional shapes.                   three- and two-dimensional shapes to
    • 2D figures – square,            identify, classify, and describe shapes.
       rectangle, circle,
       triangle

4.2.A.4 – Recognize, describe,        Pg. 217, No. 7: Explore geometric
extend and create designs and         transformations such as rotations
patterns with geometric               (turns), reflections (flips), and
objects of different shapes and       translations (slides).
colors.                               Pg. 217, No. 9: Understand the variety
                                      of ways in which geometric shapes and
                                      objects can be measured.
                                      Pg. 217, No. 10: Investigate the
                                      occurrence of geometry in nature, art,
                                      and other areas.
                                      Pg. 216, No. 6: Use tessellations to
                                      explore properties of geometric shapes
                                      and their relationships to the concepts
                                      of area and perimeter.


4.2.B. Transforming Shapes            Pgs. 338-339, Overview

4.2.B.1 – Use simple shapes to        Pg. 340, No. 1: Reproduce, extend,
make designs, patterns and            create, and describe patterns and

                                 23
pictures.                           sequences using a variety of materials.
                                    Pg. 342, No. 5: Observe and recognize
                                    examples of patterns, relationships,
                                    and functions in other disciplines and
                                    contexts.
                                    Pg. 343, No. 6: Form and verify
                                    generalizations based on observations
                                    of patterns and relationships.
                                    Pg. 216, No. 6: Use tessellations to
                                    explore properties of geometric shapes
                                    and their relationships to the concepts
                                    of area and perimeter.
                                    Pg. 217, No. 9: Understand the variety
                                    of ways in which geometric shapes and
                                    objects can be measured.
                                    Pg. 217, No. 10: Investigate the
                                    occurrence of geometry in nature, art,
                                    and other areas.



4.2.D. Units of Measurement         Pgs. 284-285, Overview

4.2.D.1 – Directly compare and      Pg. 287, No. 2: Compare and order
order objects according to          objects according to some measurable
measurable attributes.              attribute.
    • Attributes – length,          Pg. 288, No. 4: Develop and use
       weight, capacity, time,      personal referents for standard units of
       temperature                  measure (such as the width of a finger
                                    to approximate a centimeter).

4.2.D.2 – Recognize the need        Pg. 288, No. 3: Recognize the need
for a uniform unit of               for a uniform unit of measure.
measurement.


                               24
                                  4.2.D.4 – Estimate measures.          Pg. 288, No. 6: Understand and
                                                                        incorporate estimation and repeated
                                                                        measures in measurement activities.
4.3 All students will represent   4.3.A. Patterns                       Pgs. 338-339, Overview
    and analyze relationships
    among variable quantities     4.3.A.1 – Recognize, describe,        Pg. 340, No. 1: Reproduce, extend,
    and solve problems            extend, and create patterns.          create, and describe patterns and
    involving patterns,               • Using concrete                  sequences using a variety of materials.
    functions, and algebraic             materials                      Pg. 341, No. 2: Use tables, rules,
    concepts and processes.              (manipulatives),               variables, open sentences, and graphs
4.3.A. Patterns                          pictures, rhythms, and         to describe patterns and other
4.3.C. Modeling                          whole numbers.                 relationships.
                                      • Descriptions using
                                         words and symbols
                                         (e.g., “add two” or
                                         “+2”)


                                  4.3.C. Modeling                       Pg. 490, Overview

                                 4.3.C.1 – Recognize and                Pg. 491, No. 2: Investigate and
                                 describe changes over time             describe how certain quantities change
                                 (e.g., temperature, height).           over time.
4.4 All students will develop an 4.4.C. Discrete Mathematics –          Pgs. 445-447, Overview
    understanding of the         Systematic Listing and
    concepts and techniques of Counting
    data analysis, probability,
    and discrete mathematics,    4.4.C.1 – Sort and classify            Pg. 449, No. 4: Investigate ways to
    and will use them to model objects according to attributes.         represent and classify data according
    situations, solve problems,      • Venn diagrams                    to attributes, such as shape or color,
    and analyze and draw                                                and relationships, and discuss the
    appropriate inferences from                                         purpose and usefulness of such
    data.                                                               classification.
4.4.C. Discrete Mathematics –
Systematic Listing and

                                                                   25
Counting                        4.4.D. Discrete Mathematics –      Pgs. 445-447, Overview
4.4.D. Discrete Mathematics –   Vertex-Edge Graphs and
Vertex-Edge Graphs and          Algorithms
Algorithms
                                4.4.D.1 – Follow simple sets of    Pg. 450, No. 5: Follow, devise, and
                                directions (e.g., from one         describe practical lists of instructions.
                                location to another, or from a     Pg. 448, No. 2: Use networks and tree
                                recipe).                           diagrams to represent everyday
                                                                   situations.

                                4.4.D.2 – Color simple maps        Pg. 448, No. 2: Use networks and tree
                                with a small number of colors.     diagrams to represent everyday
                                                                   situations.




                                                              26
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                      27
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                      28
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             29
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       30
                                                           MATHEMATICS
                                                            FIRST GRADE

         STANDARD                     STUDENT OUTCOME                      SUGGESTED ACTIVITIES              TEACHER’S NOTES AND
                                                                            NJ FRAMEWORKS 1996                 SUPPLEMENTARY
                                                                                                                 RESOURCES
4.1 All students will develop      4.1.A. Number Sense                    Pgs. 176-177, Overview
    number sense and will
    perform standard numerical     4.1.A.1 – Use real-life                Pg. 178, No. 1: Use real-life
    operations and estimations     experiences, physical materials,       experiences, physical materials,
    on all types of numbers in a   and technology to construct            and technology to construct
    variety of ways.               meanings for numbers.                  meanings for whole numbers,
4.1.A. Number Sense                    • Whole numbers through            commonly used fractions, and
4.1.B. Numerical Operations               hundreds                        decimals.
4.1.C. Estimation                      • Ordinals
                                       • Proper fractions
                                          (denominators of 2, 3,
                                          4)

                                   4.1.A.2 – Demonstrate an               Pg. 179, No. 4: Develop a sense
                                   understanding of whole number          of the magnitudes of whole
                                   place value concepts.                  numbers, commonly used
                                                                          fractions, and decimals.

                                   4.1.A.3 – Understand that              Pg. 180, No. 5: Understand the
                                   numbers have a variety of uses.        various uses of numbers
                                                                          including counting, measuring,
                                                                          labeling, and indicating
                                                                          location.

                                   4.1.A.4 – Count and perform            Pg. 180, No. 6: Count and
                                   simple computations with               perform simple computations
                                   coins.                                 with money.
                                       • Amounts up to $1.00
                                          (using cents notation)

                                                                     31
4.1.A.5 – Compare and order        Pg. 181, No. 8: Compare and
whole numbers.                     order whole numbers,
                                   commonly used fractions, and
                                   decimals.

4.1.B. Numerical Operations        Pgs. 253-255, Overview

4.1.B.1 – Develop the meanings     Pg. 256, No. 1: Develop
of addition and subtraction by     meaning for the four basic
concretely modeling and            arithmetic operations by
discussing a large variety of      modeling and discussing a
problems.                          variety of problems.
    • Joining, separating, and     Pg. 257, No. 2: Develop
        comparing                  proficiency with and memorize
                                   basic number facts using a
                                   variety of fact strategies (such
                                   as “counting on” and
                                   “doubles”).
                                   Pg. 257, No. 3: Construct, use
                                   and explain procedures for
                                   performing whole number
                                   calculations in the various
                                   methods of computation.
                                   Pg. 258, No. 5: Use a variety of
                                   mental computation and
                                   estimation techniques.
                                   Pg. 259, No. 6: Select and use
                                   appropriate computational
                                   methods from mental math,
                                   estimation, paper-and-pencil,
                                   and calculator methods, and
                                   check, the reasonableness of
                                   results.
                                   Pg. 259, No. 7: Understand and
                                   use relationships among

                              32
                                     operations and properties of
                                     operations.

4.1.B.3 – Develop proficiency        Pg. 257, No. 2: Develop
with basic addition and              proficiency with and memorize
subtraction number facts using       basic number facts using a
a variety of fact strategies         variety of fact strategies (such
(such as “counting on” and           as “counting on” and
“near doubles”) and then             “doubles”).
commit them to memory.

4.1.B.4 – Construct, use, and        Pg. 257, No. 3: Construct, use,
explain procedures for               and explain procedures for
performing addition and              performing whole number
subtraction calculations with:       calculations in the various
    • Pencil-and-paper               methods of computation.
    • Mental math
    • Calculator

4.1.B.5 – Use efficient and          Pg. 259, No. 6: Select and use
accurate pencil-and-paper            appropriate computational
procedures for computation           methods from mental math,
with whole numbers.                  estimation, paper-and-pencil,
    • Addition of 2-digit            and calculator methods, and
       numbers                       check the reasonableness of
    • Subtraction of 2-digit         results.
       numbers

4.1.B.6 – Select pencil-and-         Pg. 259, No. 6: Select and use
paper, mental math, or a             appropriate computational
calculator as the appropriate        methods from mental math,
computational method in a            estimation, paper-and-pencil,
given situation depending on         and calculator methods, and
the context and numbers.             check the reasonableness of
                                     results.

                                33
4.1.B.7 – Check the                    Pg. 259, No. 6: Select and use
reasonableness of results of           appropriate computational
computations.                          methods from mental math,
                                       estimation, paper-and-pencil,
                                       and calculator methods, and
                                       check the reasonableness of
                                       results.
                                       Pg. 258, No. 5: Use a variety of
                                       mental computation and
                                       estimation techniques.

4.1.B.8 – Understand and use           Pg. 259, No. 7: Understand and
the inverse relationship               use relationships among
between addition and                   operations and properties of
subtraction.                           operations.


4.1.C. Estimation                      Pg. 311, Overview

4.1.C.1 – Judge without counting       Pg. 312, No. 1: Judge without
whether a set of objects has less      counting whether a set of
than, more than, or the same           objects has less than, more than,
number of objects as a reference       or the same number of objects
set.                                   as a reference set.

4.1.C.2 – Determine the                Pg. 314, No. 6: Determine the
reasonableness of an answer            reasonableness of an answer by
by estimating the result of            estimating the result of
computations (e.g., 15 + 16 is         operations.
not 211).

4.1.C.3 – Explore a variety of         Pg. 313, No. 4: Explore,
strategies for estimating both         construct, and use a variety of
quantities (e.g., the number of        estimation strategies.

                                  34
                                    marbles in a jar) and results of        Pg. 258, No. 5: Use a variety of
                                    computation.                            mental computation and
                                                                            estimation techniques.
                                                                            Pg. 259, No. 6: Select and use
                                                                            appropriate computational
                                                                            methods from mental math,
                                                                            estimation, paper-and-pencil,
                                                                            and calculator methods, and
                                                                            check the reasonableness of
                                                                            results.
                                                                            Pg. 288, No. 6: Understand and
                                                                            incorporate estimation and
                                                                            repeated measures in
                                                                            measurement activities.
4.2 All students will develop       4.2.A. Geometric Properties             Pgs. 213-214, Overview
    spatial sense and the ability
    to use geometric properties,    4.2.A.1 – Identify and describe         Pg. 215, No. 1: Explore spatial
    relationships, and              spatial relationships among             relationships such as the
    measurement to model,           objects in space and their relative     direction, orientation, and
    describe and analyze            shapes and sizes.                       perspectives of objects in space,
    phenomena.                          • Inside/outside, left/right,       their relative shapes and sizes,
4.2.A. Geometric Properties                 above/below, between            and the relations between
4.2.B. Transforming Shapes              • Smaller/larger/same size,         objects and their shadows or
4.2.C. Coordinate Geometry                  wider/narrower,                 projections.
4.2.D. Units of Measurement                 longer/shorter
4.2.E. Measuring Geometric              • Congruence (i.e. same
Objects                                     size and shape)

                                    4.2.A.2 – Use concrete objects,         Pg. 215, No. 3: Explore
                                    drawings, and computer graphics         properties of three- and two-
                                    to identify, classify, and describe     dimensional shapes using
                                    standard three-dimensional and          concrete objects, drawings, and
                                    two-dimensional shapes.                 computer graphics.
                                        • Vertex, edge, face, side          Pg. 216, No. 4: Use properties
                                        • 3D figures – cube,                of three- and two-dimensional

                                                                       35
       rectangular prism,          shapes to identify, classify, and
       sphere, cone, cylinder,     describe shapes.
       and pyramid
   •   2D figures – square,
       rectangle, circle, triangle
   •   Relationships between
       three- and two-
       dimensional shapes (i.e.,
       the face of a 3D shape is
       a 2D shape)


4.2.A.3 – Describe, identify and Pg. 215, No. 2: Explore
create instances of line         relationships among shapes,
symmetry.                        such as congruence, symmetry,
                                 similarity, and self-similarity.

4.2.A.4 – Recognize, describe,         Pg. 217, No. 7: Explore
extend and create designs and          geometric transformations such
patterns with geometric objects        as rotations (turns), reflections
of different shapes and colors.        (flips), and translations (slides).
                                       Pg. 217, No. 9: Understand the
                                       variety of ways in which
                                       geometric shapes and objects
                                       can be measured.
                                       Pg. 217, No. 10: Investigate the
                                       occurrence of geometry in
                                       nature, art, and other areas.
                                       Pg. 216, No. 6: Use
                                       tessellations to explore
                                       properties of geometric shapes
                                       and their relationships to the
                                       concepts of area and perimeter.



                                  36
4.2.B. Transforming Shapes            Pgs. 338-339, Overview

4.2.B.1 – Use simple shapes to        Pg. 340, No. 1: Reproduce,
make designs, patterns and            extend, create, and describe
pictures.                             patterns and sequences using a
                                      variety of materials.
                                      Pg. 342, No. 5: Observe and
                                      recognize examples of patterns,
                                      relationships, and functions in
                                      other disciplines and contexts.
                                      Pg. 343, No. 6: Form and verify
                                      generalizations based on
                                      observations of patterns and
                                      relationships.
                                      Pg. 216, No. 6: Use
                                      tessellations to explore
                                      properties of geometric shapes
                                      and their relationships to the
                                      concepts of area and perimeter.
                                      Pg. 217, No. 9: Understand the
                                      variety of ways in which
                                      geometric shapes and objects
                                      can be measured.
                                      Pg. 217, No. 10: Investigate the
                                      occurrence of geometry in
                                      nature, art, and other areas.

4.2.B.2 – Combine and                 Pg. 216, No. 5: Investigate and
subdivide simple shapes to            predict the results of combining,
make other shapes.                    subdividing, and changing
                                      shapes.


4.2.C. Coordinate Geometry            Pgs. 445-447, Overview


                                 37
4.2.C.1 – Give and follow          Pg. 217, No. 8: Develop the
directions for getting from one    concepts coordinates and paths,
point to another on a map or       using maps, tables, and grids.
grid.                              Pg. 450, No. 5: Follow, devise,
                                   and describe practical lists of
                                   instructions.


4.2.D. Units of Measurement        Pgs. 284-285, Overview

4.2.D.1 – Directly compare and     Pg. 287, No. 2: Compare and
order objects according to         order objects according to some
measurable attributes.             measurable attribute.
    • Attributes – length,         Pg. 288, No. 4: Develop and
       weight, capacity, time,     use personal referents for
       temperature.                standard units of measure (such
                                   as the width of a finger to
                                   approximate a centimeter).


4.2.D.2 – Recognize the need for   Pg. 288, No. 3: Recognize the
a uniform unit of measurement.     need for a uniform unit of
                                   measure.

4.2.D.3 – Select and use           Pg. 288, No. 5: Select and use
appropriate standard and non-      appropriate standard and non-
standard units of measure and      standard units of measurement
standard measurement tools to      to solve real-life problems.
solve real-life problems.
    • Length – inch, foot,
       yard, centimeter, meter
    • Weight – pound, gram,
       kilogram
    • Capacity – pint, quart,
       liter

                              38
                                     •   Time – second, minute,
                                         hour, day, week, month,
                                         year

                                     •   Temperature – degrees
                                         Celsius, degrees
                                         Fahrenheit

                                  4.2.D.4 – Estimate measures.        Pg. 288, No. 6: Understand and
                                                                      incorporate estimation and
                                                                      repeated measures in
                                                                      measurement activities.


                                  4.2.E. Measuring Geometric          Pgs. 213-214, Overview
                                  Objects

                                  4.2.E.1 – Directly measure the      Pg. 215, No. 3: Explore
                                  perimeter of simple two-            properties of three- and two-
                                  dimensional shapes.                 dimensional shapes using
                                                                      concrete objects, drawings, and
                                                                      computer graphics.
                                                                      Pg. 286, No. 1: Use and
                                                                      describe measures of length,
                                                                      distance, capacity, weight, area,
                                                                      volume, time, and temperature.

                                  4.2.E.2 – Directly measure the      Pg. 216, No. 6: Use
                                  area of simple two-dimensional      tessellations to explore
                                  shapes by covering them with        properties of geometric shapes
                                  squares.                            and their relationships to the
                                                                      concepts of area and perimeter.

4.3 All students will represent   4.3.A. Patterns                     Pgs. 338-339, Overview
    and analyze relationships

                                                                 39
    among variable quantities   4.3.A.1 – Recognize, describe,       Pg. 340, No. 1: Reproduce,
    and solve problems          extend, and create patterns.         extend, create, and describe
    involving patterns,             • Using concrete materials       patterns and sequences using a
    functions, and algebraic           (manipulatives), pictures,    variety of materials.
    concepts and processes.            rhythms, and whole            Pg. 341, No. 2: Use tables,
4.3.A Patterns                         numbers.                      rules, variables, open sentences,
4.3.B Functions and                 • Descriptions using words       and graphs to describe patterns
       Relationships                   and symbols (e.g., “add       and other relationships.
4.3.C Modeling                         two” or “+2”)
4.3.D Procedures                    • Repeating patterns
                                    • Whole-number patterns
                                       that grow or shrink as a
                                       result of repeatedly
                                       adding or subtracting a
                                       fixed number (e.g., skip
                                       counting forward or
                                       backward)

                                4.3.B. Functions and                 Pgs. 338-339, Overview
                                Relationships

                                4.3.B.1 – Use concrete and           Pg. 341, No. 3: Use concrete
                                pictorial models of function         and pictorial models to explore
                                machines to explore the basic        the basic concept of a function.
                                concept of a function.


                                4.3.C. Modeling                      Pg. 490, Overview

                                4.3.C.1 – Recognize and              Pg. 491, No. 2: Investigate and
                                describe changes over time (e.g.,    describe how certain quantities
                                temperature, height).                change over time.

                                4.3.C.2 – Construct and solve        Pgs. 408-409, Overview
                                simple open sentences                Pg. 411, No. 4: Construct and

                                                                40
                                   involving addition or              solve open sentences (examples:
                                   subtraction.                       3 + ‫ )7 = ٱ‬that describe real-life
                                      • Result unknown (e.g., 6       situations.
                                          – 2 = __ or n = 3 + 5)
                                      • Part unknown (e.g., 3 +
                                          __ = 8)


                                   4.3.D. Procedures                  Pgs. 253-255, Overview

                                   4.3.D.1 – Understand and           Pg. 259, No. 7: Understand and
                                   apply (but don’t name) the         use relationships among
                                   following properties of            operations and properties of
                                   addition:                          operations.
                                       • Commutative (e.g., 5 +
                                          3 = 3 + 5)
                                       • Zero as the identity
                                          element (e.g., 7 + 0 = 7)
                                       • Associative (e.g., 7.+ 3 +
                                          2 can be found by first
                                          adding either 7 + 3 or 3
                                          + 2)
4.4 All students will develop an   4.4.A. Data Analysis               Pgs. 445-447, Overview
    understanding of the
    concepts and techniques of     4.4.A.1 – Collect, generate,       Pg. 376, No. 1: Formulate and
    data analysis, probability,    record, and organize data in       solve problems that involve
    and discrete mathematics,      response to questions, claims,     collecting, organizing, and
    and will use them to model     or curiosity.                      analyzing data.
    situations, solve problems,        • Data collected from
    and analyze and draw                   students’ everyday
    appropriate inferences from            experiences
    data.                              • Data generated from
4.4.A. Data Analysis                       chance devices, such as
4.4.B. Probability                         spinners and dice
4.4.C. Discrete Mathematics –

                                                                 41
Systematic Listing and          4.4.A.2 – Read, interpret,          Pg. 376, No. 1: Formulate and
Counting                        construct, and analyze displays     solve problems that involve
4.4.D. Discrete Mathematics –   of data.                            collecting, organizing, and
Vertex-Edge Graphs and              • Pictures, tally chart,        analyzing data.
Algorithms                             pictograph, bar graph,       Pg. 377, No. 3: Make
                                       Venn diagram                 inferences and formulate
                                    • Smallest to largest,          hypotheses based on data.
                                       most frequent (mode)         Pg. 377, No. 4: Understand and
                                                                    informally use the concepts of
                                                                    range, mean, mode, and median.
                                                                    Pg. 377, No. 5: Construct, read,
                                                                    and interpret displays of data
                                                                    such as pictographs, bar graphs,
                                                                    circle graphs, tables, and lists.


                                4.4.B. Probability                  Pgs. 374-375, Overview

                                4.4.B.1 – Use chance devices        Pg. 376, No. 2: Generate and
                                like spinners and dice to           analyze data obtained using
                                explore concepts of                 chance devices such as spinners
                                probability.                        and dice.
                                    • Certain, impossible           Pg. 378, No. 6: Determine the
                                    • More likely, less likely,     probability of a simple event,
                                        equally likely              assuming equally likely
                                                                    outcomes.

                                4.4.B.2 – Provide probability of    Pg. 378, No. 6: Determine the
                                specific outcomes.                  probability of a simple event,
                                    • Probability of getting        assuming equally likely
                                        specific outcome when a     outcomes.
                                        coin is tossed, when die
                                        is rolled, when spinner     Pg. 378, No. 7: Make
                                        is spun (e.g., if spinner   predictions that are based on
                                        has five equal sectors,     intuitive, experimental, and

                                                               42
       then probability of              theoretical probabilities.
       getting a particular             Pg. 378, No. 8: Use concepts of
       sector is one out of five)       certainty, fairness, and chance
   •   When picking a marble            to discuss the probability of
       from a bag with three            actual events.
       red marbles and four
       blue marbles, the
       probability of getting a
       red marble is three out
       of seven.


4.4.C. Discrete Mathematics –           Pgs. 445-447, Overview
Systematic Listing and
Counting

4.4.C.1 – Sort and classify             Pg. 449, No. 4: Investigate
objects according to attributes.        ways to represent and classify
    • Venn diagrams                     data according to attributes,
                                        such as shape or color, and
                                        relationships, and discuss the
                                        purpose and usefulness of such
                                        classification.

4.4.C.2 – Generate all                  Pg. 447, No. 1: Explore a
possibilities in simple counting        variety of puzzles, games, and
situations (e.g., all outfits           counting problems.
involving two shirts and three
pants).


4.4.D. Discrete Mathematics –           Pgs. 445-447, Overview
Vertex-Edge Graphs and
Algorithms


                                   43
 4.4.D.1 – Follow simple sets of        Pg. 450, No. 5: Follow, devise,
 directions (e.g., from one             and describe practical lists of
 location to another, or from a         instructions.
 recipe).                               Pg. 448, No. 2: Use networks
                                        and tree diagrams to represent
                                        everyday situations.

 4.4.D.2 – Color simple maps            Pg. 448, No. 2: Use networks
 with a small number of colors.         and tree diagrams to represent
                                        everyday situations.

4.4.D.3 – Play simple two-person         Pg. 447, No. 1: Explore a
games (e.g., tic-tac-toe) and           variety of puzzles, games, and
informally explore the idea of          counting problems.
what the outcome should be.




                                   44
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                      45
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                      46
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             47
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       48
                                                           MATHEMATICS
                                                           SECOND GRADE

         STANDARD                     STUDENT OUTCOME                      SUGGESTED ACTIVITIES                TEACHER’S NOTES AND
                                                                            NJ FRAMEWORKS 1996                   SUPPLEMENTARY
                                                                                                                   RESOURCES
4.1 All students will develop      4.1.A. Number Sense                    Pgs. 176-177, Overview
    number sense and will
    perform standard numerical     4.1.A.1 – Use real-life                Pg. 178, No. 1: Use real-life
    operations and estimations     experiences, physical materials,       experiences, physical materials,
    on all types of numbers in a   and technology to construct            and technology to construct
    variety of ways.               meanings for numbers.                  meanings for whole numbers,
4.1.A. Number Sense                    • Whole numbers through            commonly used fractions, and
4.1.B. Numerical Operations               hundreds                        decimals.
4.1.C. Estimation                      • Ordinals
                                       • Proper fractions
                                          (denominators of 2, 3, 4,
                                          8, 10)

                                   4.1.A.2 – Demonstrate an               Pg. 179, No. 4: Develop a sense
                                   understanding of whole number          of the magnitudes of whole
                                   place value concepts.                  numbers, commonly used
                                                                          fractions, and decimals.

                                   4.1.A.3 – Understand that              Pg. 180, No. 5: Understand the
                                   numbers have a variety of uses.        various uses of numbers
                                                                          including counting, measuring,
                                                                          labeling, and indicating location.

                                   4.1.A.4 – Count and perform            Pg. 180, No. 6: Count and
                                   simple computations with coins.        perform simple computations
                                       • Amounts up to $1.00              with money.
                                          (using cents notation)



                                                                     49
4.1.A.5 – Compare and order        Pg. 181, No. 8: Compare and
whole numbers.                     order whole numbers,
                                   commonly used fractions, and
                                   decimals.


4.1.B. Numerical Operations        Pgs. 253-255, Overview

4.1.B.1 – Develop the meanings     Pg. 256, No. 1: Develop
of addition and subtraction by     meaning for the four basic
concretely modeling and            arithmetic operations by
discussing a large variety of      modeling and discussing a
problems.                          variety of problems.
    • Joining, separating, and     Pg. 257, No. 2: Develop
        comparing                  proficiency with and memorize
                                   basic number facts using a
                                   variety of fact strategies (such as
                                   “counting on” and “doubles”).
                                   Pg. 257, No. 3: Construct, use
                                   and explain procedures for
                                   performing whole number
                                   calculations in the various
                                   methods of computation.
                                   Pg. 258, No. 5: Use a variety of
                                   mental computation and
                                   estimation techniques.
                                   Pg. 259, No. 6: Select and use
                                   appropriate computational
                                   methods from mental math,
                                   estimation, paper-and-pencil,
                                   and calculator methods, and
                                   check, the reasonableness of
                                   results.
                                   Pg. 259, No. 7: Understand and
                                   use relationships among

                              50
                                      operations and properties of
                                      operations.

4.1.B.2 – Explore the meanings        Pg. 256, No. 1: Develop
of multiplication and division        meaning for the four basic
by modeling and discussing            arithmetic operations by
problems.                             modeling and discussing a
                                      variety of problems.
                                      Pg. 259, No. 7: Understand and
                                      use relationships among
                                      operations and properties of
                                      operations.

4.1.B.3 – Develop proficiency         Pg. 257, No. 2: Develop
with basic addition and               proficiency with and memorize
subtraction facts using a variety     basic number facts using a
of fact strategies (such as           variety of fact strategies (such as
“counting on” and “near               “counting on” and “doubles”).
doubles”) and then commit them
to memory.

4.1.B.4 – Construct, use, and         Pg. 257, No. 3: Construct, use,
explain procedures for                and explain procedures for
performing addition and               performing whole number
subtraction calculations with:        calculations in the various
    • Pencil-and-paper                methods of computation.
    • Mental math
    • Calculator

4.1.B.5 – Use efficient and           Pg. 259, No. 6: Select and use
accurate pencil-and-paper             appropriate computational
procedures for computation with       methods from mental math,
whole numbers.                        estimation, paper-and-pencil,
    • Addition of 2-digit             and calculator methods, and
       numbers                        check the reasonableness of

                                 51
   •   Subtraction of 2-digit        results.
       numbers

4.1.B.6 – Select pencil-and-         Pg. 259, No. 6: Select and use
paper, mental math, or a             appropriate computational
calculator as the appropriate        methods from mental math,
computational method in a given      estimation, paper-and-pencil,
situation depending on the           and calculator methods, and
context and numbers.                 check the reasonableness of
                                     results.

4.1.B.7 – Check the                  Pg. 259, No. 6: Select and use
reasonableness of results of         appropriate computational
computations.                        methods from mental math,
                                     estimation, paper-and-pencil,
                                     and calculator methods, and
                                     check the reasonableness of
                                     results.
                                     Pg. 258, No. 5: Use a variety of
                                     mental computation and
                                     estimation techniques.

4.1.B.8 – Understand and use the     Pg. 259, No. 7: Understand and
inverse relationship between         use relationships among
addition and subtraction.            operations and properties of
                                     operations.

4.1.C. Estimation                    Pg. 311, Overview

4.1.C.1 – Judge without counting     Pg. 312, No. 1: Judge without
whether a set of objects has less    counting whether a set of
than, more than, or the same         objects has less than, more than,
number of objects as a reference     or the same number of objects
set.                                 as a reference set.


                                52
                                    4.1.C.2 – Determine the                 Pg. 314, No. 6: Determine the
                                    reasonableness of an answer by          reasonableness of an answer by
                                    estimating the result of                estimating the result of
                                    computations (e.g., 15 + 16 is          operations.
                                    not 211).

                                    4.1.C.3 – Explore a variety of          Pg. 313, No. 4: Explore,
                                    strategies for estimating both          construct, and use a variety of
                                    quantities (e.g., the number of         estimation strategies.
                                    marbles in a jar) and results of        Pg. 258, No. 5: Use a variety of
                                    computation.                            mental computation and
                                                                            estimation techniques.
                                                                            Pg. 259, No. 6: Select and use
                                                                            appropriate computational
                                                                            methods from mental math,
                                                                            estimation, paper-and-pencil,
                                                                            and calculator methods, and
                                                                            check the reasonableness of
                                                                            results.
                                                                            Pg. 288, No. 6: Understand and
                                                                            incorporate estimation and
                                                                            repeated measures in
                                                                            measurement activities.

4.2. All students will develop      4.2.A. Geometric Properties             Pgs. 213-214, Overview
    spatial sense and the ability
    to use geometric properties,    4.2.A.1 – Identify and describe         Pg. 215, No. 1: Explore spatial
    relationships, and              spatial relationships among             relationships such as the
    measurement to model,           objects in space and their relative     direction, orientation, and
    describe and analyze            shapes and sizes.                       perspectives of objects in space,
    phenomena.                          • Inside/outside, left/right,       their relative shapes and sizes,
4.2.A. Geometric Properties                 above/below, between            and the relations between
4.2.B. Transforming Shapes              • Smaller/larger/same size,         objects and their shadows or
4.2.C. Coordinate Geometry                  wider/narrower,                 projections.
4.2.D. Units of Measurement                 longer/shorter

                                                                       53
4.2.E. Measuring Geometric      •   Congruence (i.e. same
Objects                             size and shape)

                             4.2.A.2 – Use concrete objects,        Pg. 215, No. 3: Explore
                             drawings, and computer graphics        properties of three- and two-
                             to identify, classify, and describe    dimensional shapes using
                             standard three-dimensional and         concrete objects, drawings, and
                             two-dimensional shapes.                computer graphics.
                                 • Vertex, edge, face, side         Pg. 216, No. 4: Use properties
                                 • 3D figures – cube,               of three- and two-dimensional
                                     rectangular prism, sphere,     shapes to identify, classify, and
                                     cone, cylinder, and            describe shapes.
                                     pyramid
                                 • 2D figures – square,
                                     rectangle, circle, triangle
                                 • Relationships between
                                     three- and two-
                                     dimensional shapes (i.e.,
                                     the face of a 3D shape is
                                     a 2D shape)


                             4.2.A.3 – Describe, identify and       Pg. 215, No. 2: Explore
                             create instances of line               relationships among shapes,
                             symmetry.                              such as congruence, symmetry,
                                                                    similarity, and self-similarity.

                             4.2.A.4 – Recognize, describe,         Pg. 217, No. 7: Explore
                             extend and create designs and          geometric transformations such
                             patterns with geometric objects        as rotations (turns), reflections
                             of different shapes and colors.        (flips), and translations (slides).
                                                                    Pg. 217, No. 9: Understand the
                                                                    variety of ways in which
                                                                    geometric shapes and objects
                                                                    can be measured.

                                                               54
                                      Pg. 217, No. 10: Investigate the
                                      occurrence of geometry in
                                      nature, art, and other areas.
                                      Pg. 216, No. 6: Use
                                      tessellations to explore
                                      properties of geometric shapes
                                      and their relationships to the
                                      concepts of area and perimeter.


4.2.B. Transforming Shapes            Pgs. 338-339, Overview

4.2.B.1 – Use simple shapes to        Pg. 340, No. 1: Reproduce,
make designs, patterns and            extend, create, and describe
pictures.                             patterns and sequences using a
                                      variety of materials.
                                      Pg. 342, No. 5: Observe and
                                      recognize examples of patterns,
                                      relationships, and functions in
                                      other disciplines and contexts.
                                      Pg. 343, No. 6: Form and verify
                                      generalizations based on
                                      observations of patterns and
                                      relationships.
                                      Pg. 216, No. 6: Use
                                      tessellations to explore
                                      properties of geometric shapes
                                      and their relationships to the
                                      concepts of area and perimeter.
                                      Pg. 217, No. 9: Understand the
                                      variety of ways in which
                                      geometric shapes and objects
                                      can be measured.
                                      Pg. 217, No. 10: Investigate the
                                      occurrence of geometry in

                                 55
                                       nature, art, and other areas.

4.2.B.2 – Combine and                  Pg. 216, No. 5: Investigate and
subdivide simple shapes to make        predict the results of combining,
other shapes.                          subdividing, and changing
                                       shapes.


4.2.C. Coordinate Geometry             Pgs. 445-447, Overview

4.2.C.1 – Give and follow              Pg. 217, No. 8: Develop the
directions for getting from one        concepts coordinates and paths,
point to another on a map or           using maps, tables, and grids.
grid.                                  Pg. 450, No. 5: Follow, devise,
                                       and describe practical lists of
                                       instructions.


4.2.D. Units of Measurement            Pgs. 284-285, Overview

4.2.D.1 – Directly compare and         Pg. 287, No. 2: Compare and
order objects according to             order objects according to some
measurable attributes.                 measurable attribute.
    • Attributes – length,             Pg. 288, No. 4: Develop and
       weight, capacity, time,         use personal referents for
       temperature.                    standard units of measure (such
                                       as the width of a finger to
                                       approximate a centimeter).

4.2.D.2 – Recognize the need for       Pg. 288, No. 3: Recognize the
a uniform unit of measurement.         need for a uniform unit of
                                       measure.

4.2.D.3 – Select and use               Pg. 288, No. 5: Select and use
appropriate standard and non-          appropriate standard and non-

                                  56
standard units of measure and         standard units of measurement
standard measurement tools to         to solve real-life problems.
solve real-life problems.
    • Length – inch, foot, yard,
        centimeter, meter
    • Weight – pound, gram,
        kilogram
    • Capacity – pint, quart,
        liter
    • Time – second, minute,
        hour, day, week, month,
        year
    • Temperature – degrees
        Celsius, degrees
        Fahrenheit


4.2.D.4 – Estimate measures.          Pg. 288, No. 6: Understand and
                                      incorporate estimation and
                                      repeated measures in
                                      measurement activities.


4.2.E. Measuring Geometric            Pgs. 213-214, Overview
Objects

4.2.E.1 – Directly measure the        Pg. 215, No. 3: Explore
perimeter of simple two-              properties of three- and two-
dimensional shapes.                   dimensional shapes using
                                      concrete objects, drawings, and
                                      computer graphics.
                                      Pg. 286, No. 1: Use and
                                      describe measures of length,
                                      distance, capacity, weight, area,
                                      volume, time, and temperature.

                                 57
                                  4.2.E.2 – Directly measure the        Pg. 216, No. 6: Use
                                  area of simple two-dimensional        tessellations to explore
                                  shapes by covering them with          properties of geometric shapes
                                  squares.                              and their relationships to the
                                                                        concepts of area and perimeter.

4.3 All students will represent   4.3.A. Patterns                       Pgs. 338-339, Overview
    and analyze relationships
    among variable quantities     4.3.A.1 – Recognize, describe,        Pg. 340, No. 1: Reproduce,
    and solve problems            extend, and create patterns.          extend, create, and describe
    involving patterns,               • Using concrete materials        patterns and sequences using a
    functions, and algebraic             (manipulatives), pictures,     variety of materials.
    concepts and processes.              rhythms, and whole             Pg. 341, No. 2: Use tables,
4.3.A. Patterns                          numbers.                       rules, variables, open sentences,
4.3.B. Functions and                  • Descriptions using words        and graphs to describe patterns
Relationships                            and symbols (e.g., “add        and other relationships.
4.3.C. Modeling                          two” or “+2”)
4.3.D. Procedures                     • Repeating patterns
                                      • Whole-number patterns
                                         that grow or shrink as a
                                         result of repeatedly
                                         adding or subtracting a
                                         fixed number (e.g., skip
                                         counting forward or
                                         backward)

                                  4.3.B. Functions and                  Pgs. 338-339, Overview
                                  Relationships

                                  4.3.B.1 – Use concrete and            Pg. 341, No. 3: Use concrete
                                  pictorial models of function          and pictorial models to explore
                                  machines to explore the basic         the basic concept of a function.
                                  concept of a function.



                                                                   58
                                   4.3.C. Modeling                      Pg. 490, Overview

                                   4.3.C.1 – Recognize and              Pg. 491, No. 2: Investigate and
                                   describe changes over time (e.g.,    describe how certain quantities
                                   temperature, height).                change over time.

                                   4.3.C.2 – Construct and solve        Pgs. 408-409, Overview
                                   simple open sentences involving      Pg. 411, No. 4: Construct and
                                   addition or subtraction.             solve open sentences (examples:
                                       • Result unknown (e.g., 6 –      3 + ‫ )7 = ٱ‬that describe real-life
                                          2 = __ or n = 3 + 5)          situations.
                                       • Part unknown (e.g., 3 +
                                          __ = 8)

                                   4.3.D. Procedures                    Pgs. 253-255, Overview

                                   4.3.D.1 – Understand and apply       Pg. 259, No. 7: Understand and
                                   (but don’t name) the following       use relationships among
                                   properties of addition:              operations and properties of
                                       • Commutative (e.g., 5 + 3       operations.
                                          = 3 + 5)
                                       • Zero as the identity
                                          element (e.g., 7 + 0 = 7)
                                       • Associative (e.g., 7.+ 3 +
                                          2 can be found by first
                                          adding either 7 + 3 or 3 +
                                          2)

4.4 All students will develop an   4.4.A. Data Analysis                 Pgs. 445-447, Overview
    understanding of the
    concepts and techniques of     4.4.A.1 – Collect, generate,         Pg. 376, No. 1: Formulate and
    data analysis, probability,    record, and organize data in         solve problems that involve
    and discrete mathematics,      response to questions, claims, or    collecting, organizing, and
    and will use them to model     curiosity.                           analyzing data.
    situations, solve problems,        • Data collected from

                                                                   59
    and analyze and draw                  students’ everyday
    appropriate inferences from           experiences
    data.                             •   Data generated from
4.4.A. Data Analysis                      chance devices, such as
4.4.B. Probability                        spinners and dice
4.4.C. Discrete Mathematics –
Systematic Listing and            4.4.A.2 – Read, interpret,           Pg. 376, No. 1: Formulate and
Counting                          construct, and analyze displays      solve problems that involve
4.4.D. Discrete Mathematics –     of data.                             collecting, organizing, and
Vertex-Edge Graphs and                • Pictures, tally chart,         analyzing data.
Algorithms                                pictograph, bar graph,       Pg. 377, No. 3: Make
                                          Venn diagram                 inferences and formulate
                                      • Smallest to largest, most      hypotheses based on data.
                                          frequent (mode)              Pg. 377, No. 4: Understand and
                                                                       informally use the concepts of
                                                                       range, mean, mode, and median.
                                                                       Pg. 377, No. 5: Construct, read,
                                                                       and interpret displays of data
                                                                       such as pictographs, bar graphs,
                                                                       circle graphs, tables, and lists.


                                  4.4.B. Probability                   Pgs. 374-375, Overview

                                  4.4.B.1 – Use chance devices         Pg. 376, No. 2: Generate and
                                  like spinners and dice to explore    analyze data obtained using
                                  concepts of probability.             chance devices such as spinners
                                      • Certain, impossible            and dice.
                                      • More likely, less likely,      Pg. 378, No. 6: Determine the
                                          equally likely               probability of a simple event,
                                                                       assuming equally likely
                                                                       outcomes.


                                  4.4.B.2 – Provide probability of     Pg. 378, No. 6: Determine the

                                                                  60
specific outcomes.                      probability of a simple event,
   • Probability of getting             assuming equally likely
        specific outcome when a         outcomes.
        coin is tossed, when die        Pg. 378, No. 7: Make
        is rolled, when spinner is      predictions that are based on
        spun (e.g., if spinner has      intuitive, experimental, and
        five equal sectors, then        theoretical probabilities.
        probability of getting a        Pg. 378, No. 8: Use concepts of
        particular sector is one        certainty, fairness, and chance to
        out of five)                    discuss the probability of actual
   • When picking a marble              events.
        from a bag with three red
        marbles and four blue
        marbles, the probability
        of getting a red marble is
        three out of seven.


4.4.C. Discrete Mathematics –           Pgs. 445-447, Overview
Systematic Listing and
Counting

4.4.C.1 – Sort and classify             Pg. 449, No. 4: Investigate
objects according to attributes.        ways to represent and classify
    • Venn diagrams                     data according to attributes,
                                        such as shape or color, and
                                        relationships, and discuss the
                                        purpose and usefulness of such
                                        classification.

4.4.C.2 – Generate all                  Pg. 447, No. 1: Explore a
possibilities in simple counting        variety of puzzles, games, and
situations (e.g., all outfits           counting problems.
involving two shirts and three
pants).

                                   61
4.4.D. Discrete Mathematics –           Pgs. 445-447, Overview
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Follow simple sets of         Pg. 450, No. 5: Follow, devise,
directions (e.g., from one              and describe practical lists of
location to another, or from a          instructions.
recipe).                                Pg. 448, No. 2: Use networks
                                        and tree diagrams to represent
                                        everyday situations.

4.4.D.2 – Color simple maps             Pg. 448, No. 2: Use networks
with a small number of colors.          and tree diagrams to represent
                                        everyday situations.

4.4.D.3 – Play simple two-               Pg. 447, No. 1: Explore a
person games (e.g., tic-tac-toe)        variety of puzzles, games, and
and informally explore the idea         counting problems.
of what the outcome should be.

                                        Pg. 448, No. 2: Use networks
4.4.D.4 – Explore concrete              and tree diagrams to represent
models of vertex-edge graphs            everyday situations.
(e.g., vertices as “islands” and
edges as “bridges”).
    • Paths from one vertex
         to another




                                   62
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                      63
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                      64
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             65
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       66
                                                            MATHEMATICS
                                                              GRADE 3

         STANDARD                      STUDENT OUTCOME                        SUGGESTED ACTIVITIES              TEACHER’S NOTES AND
                                                                               NJ FRAMEWORKS 1996                 SUPPLEMENTARY
                                                                                                                    RESOURCES
4.5 All students will develop      4.1.A. Number Sense                       Pgs. 183-184, Overview
    number sense and will
    perform standard numerical     4.1.A.1 – Use real-life                   Pg. 185, No. 1: Use real-life
    operations and estimations     experiences, physical materials,          experiences, physical materials,
    on all types of numbers in a   and technology to construct               and technology to construct
    variety of ways.               meanings for numbers (unless              meanings for whole numbers,
4.1.A. Number Sense                otherwise noted, all indicators           commonly used fractions, and
4.1.B. Numerical Operations        for grade 3 pertain to these sets         decimals.
4.1.C. Estimation                  of numbers as well).                      Pg. 186, No. 4: Develop a sense
                                       • Whole numbers through               of the magnitudes of whole
                                          hundred thousands                  numbers, commonly used
                                       • Commonly used fractions             fractions, and decimals.
                                          (denominators of 2, 3, 4,          Pg. 187, No. 7: Use models to
                                          5, 6, 8, 10) as part of a          relate whole numbers, commonly
                                          whole, as a subset of a            used fractions, and decimals to
                                          set, and as a location on a        each other, and to represent
                                          number line                        equivalent forms of the same
                                                                             number.

                                   4.1.A.2 – Demonstrate an                  Pg. 185, No. 2: Develop an
                                   understanding of whole number             understanding of place value
                                   place value concepts                      concepts and numeration in
                                       • Standard, expanded, and             relationship to counting and
                                          written form.                      grouping.




                                                                        67
4.1.A.3 – Identify whether any
whole number is odd or even.

4.1.A.4 – Explore the extension           Pg. 186, No. 4: Develop a sense
of the place value system to              of the magnitudes of whole
decimals through hundredths.              numbers, commonly used
                                          fractions, and decimals.
                                          Pg. 187, No. 7: Use models to
                                          relate whole numbers, commonly
                                          used fractions, and decimals to
                                          each other, and to represent
                                          equivalent forms of the same
                                          number.

4.1.A.5 – Understand the various          Pg. 187, No. 5: Understand the
uses of numbers.                          various uses of numbers
    • Counting, measuring,                including counting, measuring,
        labeling (e.g., numbers on        labeling, and indicating location.
        baseball uniforms)
    • Ordinal numbers

4.1.A.6 – Compare and order               Pg. 188, No. 8: Compare and
numbers.                                  order whole numbers, commonly
                                          used fractions, and decimals.

4.1.B. Numerical Operations               Pgs. 261-262, Overview

4.1.B.1 – Develop the meanings            Pg. 263, No. 1: Develop
of the four basic arithmetic              meaning for the four basic
operations by modeling and                arithmetic operations by
discussing a large variety of             modeling and discussing a
problems.                                 variety of problems.




                                     68
   •   Addition and subtraction:
       joining, separating,
       comparing
   •   Multiplication: repeated
       addition, area/array
   •   Division: repeated
       subtraction, sharing
   •   Zero as place holder


4.1.B.2 – Develop proficiency              Pg. 263, No. 2: Develop
with basic multiplication and              proficiency with and memorize
division number facts using a              basic number facts using a
variety of fact strategies (such as        variety of fact strategies (such as
“skip counting” and “repeated              “counting on” and “doubles”).
subtraction”).


4.1.B.3 – Construct, use, and              Pg. 264, No. 3: Construct, use,
explain procedures for                     and explain procedures for
performing whole number                    performing whole number
calculations with:                         calculations in the various
    • Pencil-and-paper                     methods of computation.
    • Mental math                          Pg. 266, No. 6: Select and use
                                           appropriate computational
    • Calculator
                                           methods from mental math,
                                           estimation, paper-and-pencil, and
                                           calculator methods, and check
                                           the reasonableness of results.




                                      69
4.1.B.4 – Use efficient and            Pg. 266, No. 6: Select and use
accurate pencil-and-paper              appropriate computational
procedures for computation with        methods from mental math,
whole numbers.                         estimation, paper-and-pencil, and
    • Addition of 3-digit              calculator methods, and check
       numbers                         the reasonableness of results.
    • Subtraction of 3-digit
       numbers
    • Multiplication of 2-digit
       numbers by 1-digit
       numbers


4.1.B.5 – Count and perform        Pg. 187, No. 6: Count and
simple computations with money. perform simple computations
    • Cents notation (¢)           with money
    • Understand relationship
       between penny, nickel,
       dime, quarter, half dollar,
       and dollar.

4.1.B.6 – Select pencil-and-           Pg. 265, No. 5: Use a variety of
paper, mental math, or a               mental computation and
calculator as the appropriate          estimation techniques.
computational method in a given
situation depending on the
context and numbers.

4.1.B.7 – Check the                    Pg. 319, No. 6: Determine the
reasonableness of results of           reasonableness of an answer by
computations.                          estimating the result of
                                       operations.




                                  70
4.1.C. Estimation                         Pg. 316, Overview

4.1.C.1 – Judge without counting          Pg. 317, No. 1: Judge without
whether a set of objects has less         counting whether a set of objects
than, more than, or the same              has less than, more than, or the
number of objects as a reference          same number of objects as a
set.                                      reference set.


4.1.C.2 – Construct and use a             Pg. 265, No. 5: Use a variety of
variety of estimation strategies          mental computation and
(e.g., rounding and mental math)          estimation techniques.
for estimating both quantities and        Pg. 318, No. 4: Explore,
the results of computations.              construct, and use a variety of
                                          estimation strategies.
                                          Pg. 319, No. 7: Apply estimation
                                          in working with quantities,
                                          measurement, time, computation,
                                          and problem solving.


4.1.C.3 – Recognize when an               Pg. 318, No. 5: Recognize when
estimate is appropriate, and              estimation is appropriate, and
understand the usefulness of an           understand the usefulness of an
estimate as distinct from an exact        estimate as distinct from an exact
answer.                                   answer.

4.1.C.4 – Use estimation to               Pg. 319, No. 6: Determine the
determine whether the result of a         reasonableness of an answer by
computation (either by calculator         estimating the result of
or by hand) is reasonable.                operations.




                                     71
4.6 All students will develop       4.2.A. Geometric Properties                 Pgs. 219-220, Overview
    spatial sense and the ability
    to use geometric properties,    4.2.A.1 – Identify and describe             Pg. 221, No. 1: Explore spatial
    relationships, and              spatial relationships of two or             relationships such as the
    measurement to model,           more objects in space.                      direction, orientation, and
    describe and analyze                • Direction, orientation, and           perspectives of objects in space,
    phenomena.                              perspectives (e.g., which           their relative shapes and sizes,
4.2.A. Geometric Properties                 object is on your left              and the relations between objects
4.2.B. Transforming Shapes                  when you are standing               and their shadows or projections.
4.2.C. Coordinate Geometry                  here?)
4.2.D. Units of Measurement             • Relative shapes and sizes
4.2.E. Measuring Geometric
Objects
                                    4.2.A.2 – Use properties of                 Pg. 222, No. 3: Explore
                                    standard three-dimensional and              properties of three- and two-
                                    two-dimensional shapes to                   dimensional shapes using
                                    identify, classify, and describe            concrete objects, drawings, and
                                    them.                                       computer graphics.
                                        • Vertex, edge, face, side,             Pg. 222, No. 4: Use properties of
                                            angle                               three- and two-dimensional
                                        • 3D figures – cube,                    shapes to identify , classify, and
                                            rectangular prism, sphere,          describe shapes.
                                            cone, cylinder, and                 Pg. 222, No. 5: Investigate and
                                            pyramid                             predict the results of combining,
                                        • 2D figures – square,                  subdividing, and changing
                                            rectangle, circle, triangle,        shapes.
                                            pentagon, hexagon,
                                            octagon

                                    4.2.A.3 – Identify and describe             Pg. 221, No. 2: Explore
                                    relationships among two-                    relationships among shapes, such
                                    dimensional shapes.                         as congruence, symmetry,
                                                                                similarity, and self-similarity.



                                                                           72
   •   Same size, same shape,
       congruency
   •   Lines of symmetry


4.2.A.4 – Understand and apply
concepts involving lines, angles,
and circles.
    • Line, line segment,
        endpoint

4.2.A.5 – Recognize, describe,
extend and create space-filling
patterns.


4.2.B. Transforming Shapes

4.2.B.1 – Describe and use               Pg. 223, No. 7: Explore
geometric transformations (slide,        geometric transformations such
flip, turn).                             as rotations (turns), reflections
                                         (flips), and translations (slides).


4.2.B.2 – Investigate the                Pg. 223, No. 10: Investigate the
occurrence of geometry in nature         occurrence of geometry in nature,
and art.                                 art, and other areas.
                                         Pg. 457, No. 3: Identify and
                                         investigate sequences and
                                         patterns found in nature, art, and
                                         music.




                                    73
4.2.C. Coordinate Geometry

4.2.C.1 – Locate and name points            Pg. 223, No. 8: Develop the
in the first quadrant on a                  concepts of coordinates and
coordinate grid.                            paths, using maps, tables, and
                                            grids.

4.2.D. Units of Measurement                 Pg. 290, Overview
                                            Pg. 493, Overview

4.2.D.1 – Understand that                   Pg. 223, No. 9: Understand the
everyday objects have a variety             variety of ways in which
of attributes, each of which can            geometric shapes and objects can
be measured in many ways.                   be measured.

4.2.D.2 – Select and use                    Pg. 292, No. 2: Compare and
appropriate standard units of               order objects according to some
measure and measurement tools               measurable attribute.
to solve real-life problems.                Pg. 292, No. 5: Select and use
    • Length – fractions of an              appropriate standard and non-
        inch (1/4, 1/2), foot, mile,        standard units of measurement to
        decimeter, kilometer,               solve real-life problems.
        meter                               Pg. 495, No. 3: Experiment with
    • Area – square inch,                   approximating length, area, and
        square centimeter                   volume, using informal
    • Weight – ounce, pound,                measurement instruments.
        gram, kilogram                      Pg. 291, No. 1: Use and describe
    • Capacity – fluid ounce,               measures of length, distance,
        cup, pint, quart, gallon,           capacity, weight, area, volume,
        millimeter, liter                   time, and temperature.




                                       74
                                     4.2.D.3 – Incorporate estimation        Pg. 317, No. 3: Visually estimate
                                     in measurement activities (e.g.,        length, area, volume, or angle
                                     estimate before measuring).             measure.
                                                                             Pg. 293, No. 6: Understand and
                                                                             incorporate estimation and
                                                                             repeated measures in
                                                                             measurement activities.

                                     4.2.E. Measuring Geometric              Pg. 290, Overview
                                     Objects

                                     4.2.E.1 – Determine the area of         Pg. 291, No. 1: Use and describe
                                     simple two-dimensional shapes           measures of length, distance,
                                     on a square grid.                       capacity, weight, area, volume,
                                                                             time, and temperature.

                                     4.2.E.2 – Determine the                 Pg. 291, No. 1: Use and describe
                                     perimeter of simple shapes by           measures of length, distance,
                                     measuring all of the sides.             capacity, weight, area, volume,
                                                                             time, and temperature.

                                     4.2.E.3 – Measure and compare           Pg. 291, No. 1: Use and describe
                                     the volume of three-dimensional         measures of length, distance,
                                     objects using materials such as         capacity, weight, area, volume,
                                     rice or cubes.                          time, and temperature.
4.7 All students will represent      4.3.A. Patterns                         Pg. 345, Overview
    and analyze relationships
    among variable quantities        4.3.A.1 – Recognize, describe,          Pg. 346, No. 1: Reproduce,
    and solve problems               extend, and create patterns.            extend, create, and describe
    involving patterns, functions,       • Descriptions using words          patterns and sequences using a
    and algebraic concepts and              and number                       variety of materials.
    processes.                              sentences/expressions




                                                                        75
4.3.A. Patterns           •   Whole number patterns            Pg. 494, No. 2: Investigate and
4.3.B. Functions and          that grow or shrink as a         describe how certain quantities
Relationships                 result of repeatedly             change over time.
4.3.C. Modeling               adding, subtracting,
4.3.D. Procedures             multiplying by, or
                              dividing by a fixed
                              number (e.g., 5, 8,11 …or
                              800, 400, 200, …)


                       4.3.B. Functions and                    Pgs. 413-414, Overview
                       Relationships

                       4.3.B.1 – Use concrete and              Pg. 347, No. 2: Use tables, rules,
                       pictorial models to explore the         variables, open sentences, and
                       basic concept of a function.            graphs to describe patterns and
                           • Input/output tables, T-           other relationships.
                               charts                          Pg. 347, No. 3: Use concrete and
                                                               pictorial models to explore the
                                                               basic concept of a function.


                       4.3.C. Modeling

                       4.3.C.1 – Recognize and describe        Pg. 348, No. 4: Observe and
                       change in quantities.                   explain how a change in one
                           • Graphs representing               physical quantity can produce a
                              change over time (e.g.,          corresponding change in another.
                              temperature, height)




                                                          76
4.3.C.2 – Construct and solve              Pg. 415, No. 1: Understand and
simple open sentences involving            represent numerical situations
addition or subtraction (e.g., 3 +         using variables, expressions, and
6 = ___, n = 15 – 3, 3 + ___ = 3,          number sentences.
16 – c = 7).                               Pg. 417, No. 4: Construct and
                                           solve open sentences (example:
                                           3 + ‫ )7 = ٱ‬that describe real-life
                                           situations.

                                           Pg. 348, No. 5: Observe and
                                           recognize examples of patterns,
                                           relationships, and functions in
                                           other disciplines and contexts.
                                           Pg. 415, No. 2: Represent
                                           situations and number patterns
                                           with concrete materials, tables,
                                           graphs, verbal rules, and number
                                           sentences, and translate from one
                                           to another.

                                           Pgs. 261-262, Overview
4.3.D. Procedures
                                           Pg. 416, No. 3: Understand and
4.3.D.1 – Understand and apply             use properties of operations and
the properties of operations and           numbers.
numbers.                                   Pg. 266, No. 7: Understand and
    • Commutative (e.g., 3 X 7             use relationships among
       = 7 X 3)                            operations and properties of
    • Identify element for                 operations.
       multiplication is 1 (e.g., 1
       X 8 = 8)




                                      77
                                      •   Any number multiplied
                                          by zero is zero

                                   4.3.D.2 – Understand and use the
                                   concepts of equals, less than, and
                                   greater than to describe relations
                                   between numbers.
                                       • Symbols (=, <, >)

4.8 All students will develop an   4.4.A. Data Analysis                      Pgs. 380-381, Overview
    understanding of the
    concepts and techniques of     4.4.A.1 – Collect, generate,              Pg. 382, No. 1: Formulate and
    data analysis, probability,    organize, and display data in             solve problems that involve
    and discrete mathematics,      response to questions, claims, or         collecting, organizing, and
    and will use them to model     curiosity.                                analyzing data.
    situations, solve problems,        • Data collected from the
    and analyze and draw                   classroom environment
    appropriate inferences from
    data.                          4.4.A.2 – Read, interpret,                Pg. 383, No. 5: Construct, read,
4.4.A. Data Analysis               construct, analyze, generate              and interpret displays of data
4.4.B. Probability                 questions about, and draw                 such as pictographs, bar graphs,
4.4.C. Discrete Mathematics –      inferences from displays of data.         circle graphs, tables, and lists.
Systematic Listing and Counting        • Pictograph, bar graph,
4.4.D. Discrete Mathematics –             table
Vertex-Edge Graphs and
Algorithms
                                   4.4.B. Probability

                                   4.4.B.1 – Use everyday events             Pg. 382, No. 2: Generate and
                                   and chance devices, such as dice,         analyze data obtained using
                                   coins, and unevenly divided               chance devices such as spinners
                                   spinners, to explore concepts of          and dice.




                                                                        78
probability.                                Pg. 384, No. 6: Determine the
   • Likely, unlikely, certain,             probability of a simple event
       impossible                           assuming equally likely
   • More likely, less likely,              outcomes.
       equally likely

4.4.B.2 – Predict probabilities in          Pg. 384, No. 7: Make predictions
a variety of situations (e.g., given        that are based on intuitive,
the number of items of each color           experimental, and theoretical
in a bag, what is the probability           probabilities.
that an item picked will have a
particular color).
    • What students think will
        happen (intuitive)
    • Collect data and use that
        data to predict the
        probability (experimental)


4.4.C. Discrete Mathematics –               Pgs. 452-453, Overview
Systematic Listing and
Counting

4.4.C.1 – Represent and classify            Pg. 458, No. 4: Investigate ways
data according to attributes, such          to represent and classify data
as shape or color, and                      according to attributes, such as
relationships.                              shape or color, and relationships,
    • Venn diagrams                         and discuss the purpose and
    • Numerical and                         usefulness of such classification.
        alphabetical order




                                       79
4.4.C.2 – Represent all             Pg. 454, No. 1: Explore a variety
possibilities for a simple counting of puzzles, games, and counting
situation in an organized way and problems.
draw conclusions from this
representation.
    • Organized lists, charts

4.4.D. Discrete Mathematics –
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Follow, devise, and           Pg. 458, No. 5: Follow, devise,
describe practical sets of              and describe practical lists of
directions (e.g., to add two 2-         instructions.
digit numbers).

4.4.D.2 – Explore vertex-edge
graphs
    • Vertex, edge
    • Path

4.4.D.3 – Find the smallest             Pg. 456, No. 2: Use networks
number of colors needed to color        and tree diagrams to represent
a map.                                  everyday situations.




                                   80
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                      81
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                      82
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             83
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       84
                                                             MATHEMATICS
                                                               GRADE 4

         STANDARD                       STUDENT OUTCOME                        SUGGESTED ACTIVITIES              TEACHER’S NOTES AND
                                                                                NJ FRAMEWORKS 1996                 SUPPLEMENTARY
                                                                                                                     RESOURCES
4.1 All students will develop       4.1.A. Number Sense                       Pgs. 183-184, Overview
    number sense and will
    perform standard numerical      4.1.A.1 – Use real-life                   Pg. 185, No. 1: Use real-life
    operations and estimations on   experiences, physical materials,          experiences, physical materials,
    all types of numbers in a       and technology to construct               and technology to construct
    variety of ways.                meanings for numbers (unless              meanings for whole numbers,
4.1.A. Number Sense                 otherwise noted, all indicators           commonly used fractions, and
4.1.B. Numerical Operations         for grade 4 pertain to these sets         decimals.
4.1.C. Estimation                   of numbers as well).                      Pg. 186, No. 4: Develop a sense
                                        • Whole numbers through               of the magnitudes of whole
                                           millions                           numbers, commonly used
                                        • Apply knowledge of                  fractions, and decimals.
                                           odd/evens
                                        • Commonly used fractions
                                           (denominators of 2, 3, 4,
                                           5, 6, 8, 10, 12, and 16) as
                                           part of a whole, as a
                                           subset of a set, and as a
                                           location on a number line
                                        • Read and write mixed
                                           numbers
                                        • Decimals through
                                           hundredths, i.e., money




                                                                         85
4.1.A.2 – Demonstrate an                  Pg. 185, No. 2: Develop an
understanding of place value              understanding of place value
concepts.                                 concepts and numeration in
                                          relationship to counting and
                                          grouping.

4.1.A.3 – Demonstrate a sense of
the relative magnitudes of
numbers
    • Standard, expanded, and
        written form.

4.1.A.4 – Understand the various          Pg. 187, No. 5: Understand the
uses of numbers.                          various uses of numbers
    • Counting, measuring,                including counting, measuring,
        labeling (e.g., numbers on        labeling, and indicating location.
        baseball uniforms),
        locating (e.g., Room 235
        is on the second floor)


4.1.A.5 – Use concrete and                Pg. 187, No. 7: Use models to
pictorial models to relate whole          relate whole numbers, commonly
numbers, commonly used                    used fractions, and decimals to
fractions, and decimals to each           each other, and to represent
other, and to represent equivalent        equivalent forms of the same
forms of the same number.                 number.


4.1.A.6 – Compare and order               Pg. 188, No. 8: Compare and
numbers.                                  order whole numbers, commonly
                                          used fractions, and decimals.




                                     86
4.1.A.7 – Explore settings that            Pg. 188, No. 9: Explore real-life
give rise to negative numbers.             settings which give rise to
    • Temperatures below 0º,               negative numbers.
        debts
    • Extension of the number
        line


4.1.B. Numerical Operations                Pgs. 261-262, Overview

4.1.B.1 – Develop the meanings             Pg. 263, No. 1: Develop
of the four basic arithmetic               meaning for the four basic
operations by modeling and                 arithmetic operations by
discussing a large variety of              modeling and discussing a
problems.                                  variety of problems.
    • Addition and subtraction:
        joining, separating,
        comparing
    • Multiplication: repeated
        addition, area/array
    • Division: repeated
        subtraction, sharing


4.1.B.2 – Develop proficiency              Pg. 263, No. 2: Develop
with basic multiplication and              proficiency with and memorize
division number facts using a              basic number facts using a
variety of fact strategies (such as        variety of fact strategies (such as
“skip counting” and “repeated              “counting on” and “doubles”).
subtraction”) and then commit
them to memory.




                                      87
4.1.B.3 – Construct, use, and            Pg. 264, No. 3: Construct, use,
explain procedures for                   and explain procedures for
performing whole number                  performing whole number
calculations with:                       calculations in the various
    • Pencil-and-paper                   methods of computation.
    • Mental math
    • Calculator

4.1.B.4 – Use efficient and              Pg. 266, No. 6: Select and use
accurate pencil-and-paper                appropriate computational
procedures for computation with          methods from mental math,
whole numbers.                           estimation, paper-and-pencil, and
    • Addition of 3-digit                calculator methods, and check
       numbers                           the reasonableness of results.
    • Subtraction of 3-digit
       numbers
    • Multiplication of 2-digit
       numbers
    • Division of 3-digit
       numbers by 1-digit
       numbers
    • Zero as place holder

4.1.B.5 – Construct and use
procedures for performing
decimal addition and subtraction.


4.1.B.6 – Count and perform
                                 Pg. 187, No. 6: Count and
simple computations with money.
                                 perform simple computations
    • Standard dollars and cents with money
       notation




                                    88
4.1.B.7 – Select pencil-and-             Pg. 265, No. 5: Use a variety of
paper, mental math, or a                 mental computation and
calculator as the appropriate            estimation techniques.
computational method in a given          Pg. 266, No. 6: Select and use
situation depending on the               appropriate computational
context and numbers.                     methods from mental math,
                                         estimation, paper-and-pencil, and
                                         calculator methods, and check
                                         the reasonableness of results.

4.1.B.8 – Check the                      Pg. 319, No. 6: Determine the
reasonableness of results of             reasonableness of an answer by
computations.                            estimating the result of
                                         operations.

4.1.B.9 – Use concrete models to         Pg. 265, No. 4: Use models to
explore addition and subtraction         explore operations with fractions
with fractions.                          and decimals.

4.1.B.10 – Understand and use            Pg. 266, No. 7: Understand and
the inverse relationships between        use relationships among
addition and subtraction and             operations and properties of
between multiplication and               operations.
division.

4.1.C. Estimation                        Pg. 316, Overview

4.1.C.1 – Judge without counting         Pg. 317, No. 1: Judge without
whether a set of objects has less        counting whether a set of objects
than, more than, or the same             has less than, more than, or the
number of objects as a reference         same number of objects as a
set.                                     reference set.



                                    89
                                    4.1.C.2 – Construct and use a             Pg. 265, No. 4: Use models to
                                    variety of estimation strategies          explore operations with fractions
                                    (e.g., rounding and mental math)          and decimals.
                                    for estimating both quantities and        Pg. 265, No. 5: Use a variety of
                                    the results of computations.              mental computation and
                                                                              estimation techniques.
                                                                              Pg. 319, No. 7: Apply estimation
                                                                              in working with quantities,
                                                                              measurement, time, computation,
                                                                              and problem solving.

                                    4.1.C.3 – Recognize when an               Pg. 318, No. 5: Recognize when
                                    estimate is appropriate, and              estimation is appropriate, and
                                    understand the usefulness of an           understand the usefulness of an
                                    estimate as distinct from an exact        estimate as distinct from an exact
                                    answer.                                   answer.

                                    4.1.C.4 – Use estimation to               Pg. 319, No. 6: Determine the
                                    determine whether the result of a         reasonableness of an answer by
                                    computation (either by calculator         estimating the result of
                                    or by hand) is reasonable.                operations.
4.2 All students will develop       4.2.A. Geometric Properties               Pgs. 219-220, Overview
    spatial sense and the ability to
    use geometric properties,        4.2.A.1 – Identify and describe          Pg. 221, No. 1: Explore spatial
    relationships, and               spatial relationships of two or          relationships such as the
    measurement to model,            more objects in space.                   direction, orientation, and
    describe and analyze                 • Direction, orientation, and        perspectives of objects in space,
    phenomena.                               perspectives (e.g., which        their relative shapes and sizes,
4.2.A. Geometric Properties                  object is on your left           and the relations between objects
4.2.B. Transforming Shapes                   when you are standing            and their shadows or projections.
4.2.C. Coordinate Geometry                   here?)
4.2.D. Units of Measurement



                                                                         90
4.2.E. Measuring Geometric
Objects                          •   Relative shapes and sizes
                                 •   Shadows (projections) of
                                     everyday objects

                             4.2.A.2 – Use properties of                 Pg. 222, No. 3: Explore
                             standard three-dimensional and              properties of three- and two-
                             two-dimensional shapes to                   dimensional shapes using
                             identify, classify, and describe            concrete objects, drawings, and
                             them.                                       computer graphics.
                                 • Vertex, edge, face, side,             Pg. 222, No. 4: Use properties of
                                     angle                               three- and two-dimensional
                                 • 3D figures – cube,                    shapes to identify, classify, and
                                     rectangular prism, sphere,          describe shapes.
                                     cone, cylinder, and                 Pg. 222, No. 5: Investigate and
                                     pyramid                             predict the results of combining,
                                 • 2D figures – square,                  subdividing, and changing
                                     rectangle, circle, triangle,        shapes.
                                     quadrilateral, pentagon,
                                     hexagon, octagon
                                 • Inclusive relationships –
                                     squares are rectangles,
                                     cubes are rectangular
                                     prisms

                             4.2.A.3 – Identify and describe             Pg. 221, No. 2: Explore
                             relationships among two-                    relationships among shapes, such
                             dimensional shapes.                         as congruence, symmetry,
                                                                         similarity, and self-similarity.
                                 • Congruence
                                 • Lines of symmetry
                                 • Similarity




                                                                    91
4.2.A.4 – Understand and apply
concepts involving lines, angles,
and circles.
    • Point, line, line segment,
        endpoint
    • Parallel, Perpendicular
    • Angles – acute, right,
        obtuse
    • Circles – diameter, radius,
        center

4.2.A.5 – Recognize, describe,
extend and create space-filling
patterns.

4.2.B. Transforming Shapes

4.2.B.1 – Use simple shapes to           Pg. 222, No. 6: Use tessellations
cover an area (tessellations).           to explore properties of
                                         geometric shapes and their
                                         relationships to the concepts of
                                         area and perimeter.

4.2.B.2 – Describe and use               Pg. 223, No. 7: Explore
geometric transformations (slide,        geometric transformations such
flip, turn).                             as rotations (turns), reflections
                                         (flips), and translations (slides).

4.2.B.3 – Investigate the                Pg. 223, No. 10: Investigate the
occurrence of geometry in nature         occurrence of geometry in nature,
and art.                                 art, and other areas.
                                         Pg. 457, No. 3: Identify and
                                         investigate sequences and



                                    92
                                           patterns found in nature, art, and
                                           music.


4.2.C. Coordinate Geometry

4.2.C.1 – Locate and name points           Pg. 223, No. 8: Develop the
in the first quadrant on a                 concepts of coordinates and
coordinate grid.                           paths, using maps, tables, and
                                           grids.

4.2.C.2 – Use coordinates to give          Pg. 223, No. 8: Develop the
or follow directions from one              concepts of coordinates and
point to another on a map or grid.         paths, using maps, tables, and
                                           grids.

4.2.D. Units of Measurement                Pg. 290, Overview
                                           Pg. 493, Overview

4.2.D.1 – Understand that
everyday objects have a variety
of attributes, each of which can
be measured in many ways.

4.2.D.2 – Select and use                   Pg. 495, No. 3: Experiment with
appropriate standard units of              approximating length, area, and
measure and measurement tools              volume, using informal
to solve real-life problems.               measurement instruments.
    • Length – fractions of an             Pg. 291, No. 1: Use and describe
        inch (1/8, 1/4, 1/2), mile,        measures of length, distance,
        decimeter, kilometer               capacity, weight, area, volume,
                                           time, and temperature.




                                      93
   •   Area – square inch,                Pg. 292, No. 2: Compare and
       square centimeter                  order objects according to some
   •   Volume – cubic inch,               measurable attribute.
       cubic centimeter                   Pg. 292, No. 5: Select and use
   •   Weight – ounce                     appropriate standard and non-
   •   Capacity – fluid ounce,            standard units of measurement to
       cup, gallon, millimeter            solve real-life problems.


4.2.D.3 – Develop and use                 Pg. 292, No. 4: Develop and use
personal referents to approximate         personal referents for standard
standard units of measure (e.g., a        units of measure (such as the
common paper clip is about an             width of a finger to approximate
inch long).                               a centimeter).
                                          Pg. 317, No. 2: Visually estimate
                                          length, area, volume, or angle
                                          measure.
                                          Pg. 292, No. 3: Recognize the
                                          need for a uniform unit of
                                          measure.

4.2.D.4 – Incorporate estimation          Pg. 317, No. 3: Visually estimate
in measurement activities (e.g.,          length, area, volume, or angle
estimate before measuring).               measure.
                                          Pg. 293, No. 6: Understand and
                                          incorporate estimation and
                                          repeated measures in
                                          measurement activities.

4.2.D.5 – Solve problems
involving elapsed time.




                                     94
                                  4.2.E. Measuring Geometric              Pg. 290, Overview
                                  Objects


                                  4.2.E.1 – Determine the area of         Pg. 291, No. 1: Use and describe
                                  simple two-dimensional shapes           measures of length, distance,
                                  on a square grid.                       capacity, weight, area, volume,
                                                                          time, and temperature.


                                  4.2.E.2 – Distinguish between           Pg. 222, No. 6: Use tessellations
                                  perimeter and area and use each         to explore properties of
                                  appropriately in problem-solving        geometric shapes and their
                                  situations.                             relationships to the concepts of
                                                                          area and perimeter.
                                                                          Pg. 291, No. 1: Use and describe
                                                                          measures of length, distance,
                                                                          capacity, weight, area, volume,
                                                                          time, and temperature.


                                  4.2.E.3 – Measure and compare           Pg. 291, No. 1: Use and describe
                                  the volume of three-dimensional         measures of length, distance,
                                  objects using materials such as         capacity, weight, area, volume,
                                  rice or cubes.                          time, and temperature.


4.3 All students will represent   4.3.A. Patterns                         Pg. 345, Overview
    and analyze relationships
    among variable quantities and
    solve problems involving      4.3.A.1 – Recognize, describe,          Pg. 346, No. 1: Reproduce,
    patterns, functions, and      extend, and create patterns.            extend, create, and describe



                                                                     95
    algebraic concepts and      •   Descriptions using words         patterns and sequences using a
    processes.                      and number                       variety of materials.
4.3.A. Patterns                     sentences/expressions,           Pg. 494, No. 1: Investigate and
4.3.B. Functions and                graphs, tables, variables        describe patterns that continue
Relationships                       (e.g., shape, blank, or          indefinitely.
4.3.C. Modeling                     letter)                          Pg. 494, No. 2: Investigate and
4.3.D. Procedures               •   Sequences that stop or           describe how certain quantities
                                    that continue infinitely         change over time.
                                •   Whole number patterns            Pg. 349, No. 6: Form and verify
                                    that grow or shrink as a         generalizations based on
                                    result of repeatedly             observations of patterns and
                                    adding, subtracting,             relationships.
                                    multiplying by, or
                                    dividing by a fixed
                                    number (e.g., 5, 8,11 …or
                                    800, 400, 200, …)
                                •   Sequences can often be
                                    extended in more than
                                    one way (e.g., the next
                                    term after 1, 2, 4, …
                                    could be 8, or 7, or …)
                                •   Form and verify
                                    generalizations based on
                                    observations of patterns
                                    and relationships.

                             4.3.B. Functions and                    Pgs. 413-414, Overview
                             Relationships

                             4.3.B.1 – Use concrete and              Pg. 347, No. 2: Use tables, rules,
                             pictorial models to explore the         variables, open sentences, and
                             basic concept of a function.            graphs to describe patterns and
                                                                     other relationships.



                                                                96
   •   Input/output tables, T-          Pg. 347, No. 3: Use concrete and
       charts                           pictorial models to explore the
   •   Combining two functions          basic concept of a function.
       machines
   •   Reversing a function
       machine

4.3.C. Modeling

4.3.C.1 – Recognize and describe        Pg. 348, No. 4: Observe and
change in quantities.                   explain how a change in one
    • Graphs representing               physical quantity can produce a
       change over time (e.g.,          corresponding change in another.
       temperature, height)
    • How change in one
       physical quantity can
       produce a corresponding
       change in another (e.g.,
       pitch of a sound depends
       on the rate of vibration)


4.3.C.2 – Construct and solve           Pg. 415, No. 1: Understand and
simple open sentences involving         represent numerical situations
any one operation (e.g., 3 X 6 =        using variables, expressions, and
___, n = 15 ÷ 3, 3 X ___ = 0, 16        number sentences.
– c = 7)                                Pg. 417, No. 4: Construct and
                                        solve open sentences (example:
                                        3 + ‫ )7 = ٱ‬that describe real-life
                                        situations.




                                   97
                                           Pg. 348, No. 5: Observe and
                                           recognize examples of patterns,
                                           relationships, and functions in
                                           other disciplines and contexts.


4.3.D. Procedures                          Pgs. 261-262, Overview

4.3.D.1 – Understand, name, and            Pg. 416, No. 3: Understand and
apply the properties of operations         use properties of operations and
and numbers.                               numbers.
    • Commutative (e.g., 3 X 7             Pg. 266, No. 7: Understand and
       = 7 X 3)                            use relationships among
    • Identify element for                 operations and properties of
       multiplication is 1 (e.g., 1        operations.
       X 8 = 8)
    • Associative (e.g., 2 X 4 X
       25 can be found by first
       multiplying either 2 x 4 or
       4 x 25)
    • Division by zero is
       undefined
    • Any number multiplied
       by zero is zero


4.3.D.2 – Understand and use the
concepts of equals, less than, and
greater than in simple number
sentences.
    • Symbols (=, <, >)




                                      98
4.4 All students will develop an    4.4.A. Data Analysis                       Pgs. 380-381, Overview
    understanding of the concepts
    and techniques of data          4.4.A.1 – Collect, generate,               Pg. 382, No. 1: Formulate and
    analysis, probability, and      organize, and display data in              solve problems that involve
    discrete mathematics, and       response to questions, claims, or          collecting, organizing, and
    will use them to model          curiosity.                                 analyzing data.
    situations, solve problems,         • Data collected from the
    and analyze and draw                    school environment
    appropriate inferences from
    data.                           4.4.A.2 – Read, interpret,                 Pg. 383, No. 5: Construct, read,
4.4.A. Data Analysis                construct, analyze, generate               and interpret displays of data
4.4.B. Probability                  questions about, and draw                  such as pictographs, bar graphs,
4.4.C. Discrete Mathematics –       inferences from displays of data.          circle graphs, tables, and lists.
Systematic Listing and Counting         • Pictograph, bar graph,
4.4.D. Discrete Mathematics –              line plot, line graph, table
Vertex-Edge Graphs and                  • Average (mean), most
Algorithms                                 frequent (mode), middle
                                           term (median)

                                    4.4.B. Probability                         Pgs. 380-381, Overview

                                    4.4.B.1 – Use everyday events              Pg. 382, No. 2: Generate and
                                    and chance devices, such as dice,          analyze data obtained using
                                    coins, and unevenly divided                chance devices such as spinners
                                    spinners, to explore concepts of           and dice.
                                    probability.                               Pg. 384, No. 8: Use concepts of
                                        • Likely, unlikely, certain,           certainty, fairness, and chance to
                                            impossible, improbable,            discuss the probability of actual
                                            fair, unfair                       events.
                                        • More likely, less likely,
                                            equally likely




                                                                          99
   •   Probability of tossing
       “heads” does not depend
       on outcomes of previous
       tosses


4.4.B.2 – Determine probabilities      Pg. 384, No. 6: Determine the
of simple events based on equally      probability of a simple event
likely outcomes and express them       assuming equally likely
as fractions.                          outcomes.


4.4.B.3 – Predict probabilities in     Pg. 384, No. 7: Make predictions
a variety of situations (e.g., given   that are based on intuitive,
the number of items of each color      experimental, and theoretical
in a bag, what is the probability      probabilities.
that an item picked will have a
particular color).
    • What students think will
        happen (intuitive)
    • Collect data and use that
        data to predict the
        probability (experimental)
    • Analyze all possible
        outcomes to find the
        probability (theoretical)


4.4.C. Discrete Mathematics –          Pgs. 452-453, Overview
Systematic Listing and
Counting




                                   100
4.4.C.1 – Represent and classify       Pg. 458, No. 4: Investigate ways
data according to attributes, such     to represent and classify data
as shape or color, and                 according to attributes, such as
relationships.                         shape or color, and relationships,
    • Venn diagrams                    and discuss the purpose and
    • Numerical and                    usefulness of such classification.
        alphabetical order


4.4.C.2 – Represent all             Pg. 454, No. 1: Explore a variety
possibilities for a simple counting of puzzles, games, and counting
situation in an organized way and problems.
draw conclusions from this
representation.
    • Organized lists, charts,
        tree diagrams
    • Dividing into categories,
        (e.g., to find the total
        number of rectangles in a
        grid, find the number of
        rectangles of each size
        and add the results)

4.4.D. Discrete Mathematics –
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Follow, devise, and          Pg. 458, No. 5: Follow, devise,
describe practical sets of             and describe practical lists of
directions (e.g., to add two 2-        instructions.
digit numbers).




                                     101
4.4.D.2 – Play two-person games      Pg. 454, No. 1: Explore a variety
and devise strategies for winning    of puzzles, games, and counting
the games (e.g., “make 5” where      problems.
players alternately add 1 or 2 and
the person who reaches 5, or
another designated number, is the
winner).


4.4.D.3 – Explore vertex-edge
graphs
    • Vertex, edge,
       neighboring/adjacent,
       number of neighbors
    • Path, circuit (i.e., path
       that ends at its starting
       point)

4.4.D.4 – Find the smallest          Pg. 456, No. 2: Use networks
number of colors needed to color     and tree diagrams to represent
a map or a graph.                    everyday situations.




                                   102
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                     103
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                     104
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             105
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       106
                                                             MATHEMATICS
                                                               GRADE 5

         STANDARD                       STUDENT OUTCOME                        SUGGESTED ACTIVITIES              TEACHER’S NOTES AND
                                                                                NJ FRAMEWORKS 1996                 SUPPLEMENTARY
                                                                                                                     RESOURCES
4.1 All students will develop       4.1.A. Number Sense                    Pgs. 190-191, Overview
    number sense and will
    perform standard numerical      4.1.A.1 – Use real-life                Pg. 192, No. 11: Extend their
    operations and estimations on   experiences, physical materials,       understanding of the number
    all types of numbers in a       and technology to construct            system by constructing meanings
    variety of ways.                meanings for numbers (unless           for integers, rational numbers,
4.1.A. Number Sense                 otherwise noted, all indicators        percents, exponents, roots,
4.1.B. Numerical Operations         for grade 5 pertain to these sets      absolute values, and numbers
4.1.C. Estimation                   of numbers as well).                   represented in scientific notation.
                                        • Demonstrate an
                                           understanding of negative
                                           numbers in real life
                                           settings (i.e., minus Cº,
                                           below sea level, football
                                           yardage)
                                        • All fractions as part of a
                                           whole, as a subset of a
                                           set, and as a location on a
                                           number line, and as
                                           divisions of whole
                                           numbers
                                        • Ratios
                                        • All decimals




                                                                         107
4.1.A.2 – Recognize the decimal       Pg. 192, No. 10: Understand
nature of United States currency      money notations, count and
and compute with money.               compute money, and recognize
                                      the decimal nature of United
                                      States currency.

4.1.A.3 – Demonstrate a sense of      Pg. 193, No. 12: Develop
the relative magnitudes of            number sense necessary for
numbers.                              estimation.
    • Standard, expanded, and         Pg. 193, No. 13: Expand the
        written form to one           sense of magnitudes of different
        million                       number types to include integers,
                                      rational numbers, and roots.

4.1.A.4 – Use whole numbers,          Pg. 195, No. 18: Investigate the
fractions, and decimals to            relationships among fractions,
represent equivalent forms of the     decimals, and percents, and use
same number.                          all of them appropriately.
                                      Pg 324, No. 9: Use equivalent
                                      representations of numbers such
                                      as fractions, decimals, and
                                      percents to facilitate estimation.

4.1.A.5 – Develop and apply           Pg. 195, No. 17: Develop and
number theory concepts in             apply number theory concepts
problem-solving situations.           such as primes, factors, and
    • Primes, factors, multiples      multiples, in real-world and
                                      mathematical problem situations.

4.1.A.6 – Compare and order           Pg. 194, No. 15: Develop and
numbers.                              use order relations for integers
                                      and rational numbers.




                                    108
4.1.B. Numerical Operations         Pg. 268, Overview

4.1.B.1 – Recognize the             Pg. 269, No. 8: Extend student
appropriate use of each             understanding and use of
arithmetic operation in problem     arithmetic operations to fractions,
situations.                         decimals, integers, and rational
                                    numbers.

4.1.B.2 – Construct, use, and       Pg. 269, No. 6: Select and use
explain procedures for              appropriate computational
performing addition and             methods from mental math,
subtraction with fractions and      estimation, paper-and-pencil, and
decimals with:                      calculator methods, and check
    • Pencil-and-paper              the reasonableness of results.
    • Mental math
    • Calculator


4.1.B.3 – Use an efficient and      Pg. 269, No. 8: Extend their
accurate pencil-and-paper           understanding and use of
procedure for division of a 3-      arithmetic operations to fractions,
digit number by a 2-digit number.   decimals, integers, and rational
                                    numbers.


4.1.B.4 – Select pencil-and-        Pg. 269, No. 6: Select and use
paper, mental math, or a            appropriate computational
calculator as the appropriate       methods from mental math,
computational method in a given     estimation, paper-and-pencil, and
situation depending on the          calculator methods, and check
context and numbers.                the reasonableness of results.




                                  109
4.1.B.5 – Check the                    Pg. 269, No. 6: Select and use
reasonableness of results of           appropriate computational
computations.                          methods from mental math,
                                       estimation, paper-and-pencil, and
                                       calculator methods, and check
                                       the reasonableness of results.

4.1.B.6 – Understand and use the
various relationships among
operations and properties of
operations.

4.1.C. Estimation                      Pg. 321, Overview

4.1.C.1 – Use a variety of             Pg. 323, No. 8: Develop, apply,
estimation strategies for both         and explain a variety of different
number and computation.                estimation strategies in problem
                                       situations involving quantities
                                       and measurement.

4.1.C.2 – Recognize when an            Pg. 322, No. 5: Recognize when
estimate is appropriate, and           estimation is appropriate, and
understand the usefulness of an        understand the usefulness of an
estimate as distinct from an exact     estimate as distinct from an exact
answer.                                answer.


4.1.C.3 – Determine the                Pg. 322, No. 6: Determine the
reasonableness of an answer by         reasonableness of an answer by
estimating the result of               estimating the result of
operations.                            operations.




                                     110
                                       4.1.C.4 – Determine whether a         Pg. 324, No. 10: Determine
                                       given estimate is an overestimate     whether a given estimate is an
                                       or an underestimate.                  overestimate or an underestimate.

4.2 All students will develop          4.2.A. Geometric Properties           Pgs. 219-220, Overview
    spatial sense and the ability to
    use geometric properties,          4.2.A.1 – Understand and apply        Pg. 229, No. 14: Understand the
    relationships, and                 concepts involving lines and          properties of lines and planes,
    measurement to model,              angles.                               including parallel and
    describe and analyze                   • Notation for line, ray,         perpendicular lines and planes,
    phenomena.                                 angle, line segment           and intersecting lines and planes
4.2.A. Geometric Properties                • Properties of parallel,         and their angles of incidence.
4.2.B. Transforming Shapes                     perpendicular, and
4.2.C. Coordinate Geometry                     intersecting lines
4.2.D. Units of Measurement                • Sum of the measures of
4.2.E. Measuring Geometric                     the interior angles of a
Objects                                        triangle is 180º


                                       4.2.A.2 – Identify, describe,         Pg. 228, No. 13: Identify,
                                       compare, classify polygons, and       describe, compare, and classify
                                       solid figures.                        plane and solid geometric
                                           • Triangles by angles and         figures.
                                                sides
                                           • Quadrilaterals, including
                                                squares, rectangles,
                                                parallelograms,
                                                trapezoids, rhombi
                                           • Polygons by number of
                                                sides
                                           • Equilateral, equiangular,
                                                regular




                                                                           111
    •   All points equidistant
        from a given point form a
        circle

4.2.A.3 – Identify similar figures.

4.2.A.4 – Understand and apply           Pg. 228, No. 12: Understand and
the concepts of congruence and           apply the concepts of symmetry,
symmetry (line and rotational).          similarity, and congruence.

4.2.B. Transforming Shapes

4.2.B.1 – Use a translation, a           Pg. 229, No. 15: Explore the
reflection, or a rotation to map         relationships among geometric
one figure onto another                  transformations (translations,
congruent figure.                        reflections, rotations, and
                                         dilations), tessellations (tilings),
                                         and congruence and similarity.

4.2.B.2 – Recognize, identify,           Pg. 231, No. 19: Investigate,
and describe geometric                   explore, and describe the
relationships and properties as          geometry in nature and real-
they exist in nature, art, and other     world applications, using models,
real-world settings.                     manipulatives, and appropriate
                                         technology.

4.2.C. Coordinate Geometry

4.2.C.1 – Create geometric               Pg. 423, No. 9: Understand and
shapes with specified properties         use the rectangular coordinate
in the first quadrant on a               system.
coordinate grid.



                                       112
4.2.D. Units of Measurement             Pgs. 294-295, Overview
                                        Pgs. 496-497, Overview


4.2.D.1 – Select and use                Pg. 298, No. 14: Understand and
appropriate units to measure            apply measurement in their own
angles and area.                        lives and in other subject areas.
                                        Pg. 230, No. 16: Develop,
                                        understand, and apply a variety
                                        of strategies for determining
                                        perimeter, area, surface area,
                                        angle measure, and volume.


4.2.D.2 – Convert measurement           Pg. 296, No. 8: Read and
units within a system (e.g., 3 feet     interpret various scales, including
= _____ inches).                        those based on number lines and
                                        maps.


4.2.D.3 – Know approximate              Pg. 298, No. 13: Convert
equivalents between the standard        measurement units from one
and metric systems (e.g., one           form to another, and carry out
kilometer is approximately 6/10         calculations that involve various
of a mile).                             units of measurement.


4.2.D.4 – Use measurements and          Pg. 298, No. 13: Convert
estimates to describe and               measurement units from one
compare phenomena.                      form to another, and carry out
                                        calculations that involve various
                                        units of measurement.




                                      113
                                       Pg. 296, No. 7: Use estimated
                                       and actual measurements to
                                       describe and compare
                                       phenomena.

4.2.E. Measuring Geometric             Pgs. 225-226, Overview
Objects

4.2.E.1 – Use a protractor to
measure angles.

4.2.E.2 – Develop and apply            Pg. 230, No. 16: Develop,
strategies and formulas for            understand, and apply a variety
finding perimeter, area, and           of strategies for determining
volume                                 perimeter, area, surface area,
    • Square/cube                      angle measure, and volume.
    • Rectangle/rectangular            Pg. 297, No. 11: Develop
        prism                          formulas and procedures for
                                       solving problems related to
                                       measurement.

4.2.E.3 – Recognize that
rectangles with the same
perimeter do not necessarily have
the same area and vice versa.


4.2.E.4 – Develop informal ways        Pg. 299, No. 15: Understand and
of approximating the measures of       explain the impact of the change
familiar objects (e.g., use a grid     of an object’s linear dimensions
to approximate the area of the         on its perimeter, area, or volume.
bottom of one’s foot).




                                     114
4.3 All students will represent     4.3.A. Patterns                       Pgs. 350-351, Overview
    and analyze relationships                                             Pgs. 418-419, Overview
    among variable quantities and
    solve problems involving        4.3.A.1 – Recognize, describe,        Pg. 231, No. 18: Explore
    patterns, functions, and        extend, and create patterns           patterns produced by processes of
    algebraic concepts and          involving whole numbers.              geometric change, relating
    processes.                          • Descriptions using tables,      iteration, approximation, and
4.3.A. Patterns                            verbal rules, simple           fractals.
4.3.B. Functions and                       equations, and graphs.         Pg. 352, No. 7: Represent and
Relationships                                                             describe mathematical
4.3.C. Modeling                                                           relationships with tables, rules,
4.3.D. Procedures                                                         simple equations, and graphs.
                                                                          Pg. 356, No. 13: Develop,
                                                                          analyze, and explain arithmetic
                                                                          sequences.

                                    4.3.B. Functions and
                                    Relationships

                                    4.3.B.1 – Describe arithmetic      Pg. 355, No. 11: Understand and
                                    operations as functions, including describe the general behavior of
                                    combining operations and           functions.
                                    reversing them.

                                    4.3.B.2 – Graph points satisfying
                                    a function from T-charts, from
                                    verbal rules, and from simple
                                    equations.

                                    4.3.C. Modeling

                                    4.3.C.1 – Use number sentences        Pg. 355, No. 12: Use patterns,
                                    to model situations.                  relationships, and linear functions



                                                                        115
   •   Using variables to            to model situations in
       represent unknown             mathematics and in other areas.
       quantities                    Pg. 354, No. 9: Use patterns,
   •   Using concrete materials,     relationships, and functions to
       tables, graphs, verbal        model situations and to solve
       rules, algebraic              problems, in mathematics and in
       expressions/equations         other subject areas.
                                     Pg. 353, No. 8: Understand and
                                     describe the relationships among
                                     various representations of
                                     patterns and functions.


4.3.C.2 – Draw freehand sketches     Pg. 420, No. 6: Represent
of graphs that model real            situations and number patterns
phenomena and use such graphs        with concrete materials, tables,
to predict and interpret events.     graphs, verbal rules, and standard
   •   Changes over time             algebraic notation.
   •   Rates of change (e.g.,        Pg. 425, No. 13: Draw freehand
       when is plant growing         sketches of, and interpret, graphs
       slowly/rapidly, when is       which model real phenomena.
       temperature dropping          Pg. 354, No. 10: Analyze
       most rapidly/slowly)          functional relationships to
                                     explain how a change in one
                                     quantity results in a change in
                                     another.
4.3.D. Procedures

4.3.D.1 – Solve simple linear        Pg. 424, No. 10: Solve simple
equations with manipulatives and     linear equations using concrete,
informally.                          informal, and graphical methods,
                                     as well as appropriate paper-and-



                                   116
                                                                           pencil techniques.
                                       •   Whole-number
                                           coefficients only, answers
                                           also whole numbers
                                        • Variables on one side of
                                           equation
4.4 All students will develop an    4.4.A. Data Analysis                   Pgs. 350-351, Overview
    understanding of the concepts
    and techniques of data          4.4.A.1 – Collect, generate,           Pg. 388, No. 9: Generate,
    analysis, probability, and      organize, and display data.            collect, organize, and analyze
    discrete mathematics, and           • Data generated from              data and represent this data in
    will use them to model                 surveys                         tables, charts, and graphs.
    situations, solve problems,
    and analyze and draw            4.4.A.2 – Read, interpret, select,     Pg. 389, No. 11: Make
    appropriate inferences from     construct, analyze, generate           inferences and formulate and
    data.                           questions about, and draw              evaluate arguments based on data
4.4.A. Data Analysis                inferences from displays of data.      analysis and data displays.
4.4.B. Probability                      • Bar graph, line graph,           Pg. 388, No. 10: Select and use
4.4.C. Discrete Mathematics –              circle graph, table             appropriate graphical
Systematic Listing and Counting         • Range, median, and mean          representations and measures of
4.4.D. Discrete Mathematics –                                              central tendency (mean, mode,
Vertex-Edge Graphs and                                                     and median) for sets of data.
Algorithms

                                    4.4.A.3 – Respond to questions    Pg. 389, No. 12: Use lines of
                                    about data and generate their own best fit to interpolate and predict
                                    questions and hypotheses.         from data.
                                                                      Pg. 389, No. 11: Make
                                                                      inferences and formulate and
                                                                      evaluate arguments based on data
                                                                      analysis and data displays.




                                                                         117
4.4.B. Probability                     Pgs. 386-387, Overview

4.4.B.1 – Determine probabilities      Pg. 391, No. 16: Interpret
of events.                             probabilities as ratios and
    • Event, probability of an         percents.
       event
    • Probability of certain
       event is 1 and of
       impossible event is 0


4.4.B.2 – Determine probability        Pg. 389, No. 13: Determine the
using intuitive, experimental, and     probability of a compound event.
theoretical methods (e.g., using       Pg. 391, No. 16: Interpret
model of picking items of              probabilities as ratios and
different colors from a bag).          percents.
    • Given numbers of various         Pg. 390, No. 15: Use models of
        types of items in a bag,       probability to predict events
        what is the probability        based on actual data.
        that an item of one type
        will be picked
    • Given data obtained
        experimentally, what is
        the likely distribution of
        items in the bag


4.4.B.3 – Model situations             Pg. 390, No. 14: Model
involving probability using            situations involving probability,
simulations (with spinners, dice)      such as genetics, using both
and theoretical models.                simulations and theoretical
                                       models.




                                     118
4.4.C. Discrete Mathematics –            Pgs. 462-463, Overview
Systematic Listing and
Counting

4.4.C.1 – Solve counting                 Pg. 464, No. 6: Use systematic
problems and justify that all            listing, counting, and reasoning
possibilities have been                  in a variety of different contexts.
enumerated without duplication
    • Organized lists, charts,
        tree diagrams, tables

4.4.C.2 – Explore the
multiplication principle of
counting in simple situations by
representing all possibilities in an
organized way (e.g., you can
make 3 X 4 = 12 outfits using 3
shirts and 4 skirts).

4.4.D. Discrete Mathematics –
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Devise strategies for          Pg. 465, No. 7: Recognize
winning simple games (e.g., start        common discrete mathematical
with two piles of objects, each of       models, explore their properties,
two players in turn removes any          and design them for specific
number of objects from a single          situations.
pile, and the person to take the
last group of objects wins) and
express those strategies as sets of
directions.




                                       119
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                     120
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                     121
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             122
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       123
                                                           MATHEMATICS
                                                             GRADE 6


         STANDARD                      STUDENT OUTCOME                       SUGGESTED ACTIVITIES              TEACHER’S NOTES AND
                                                                              NJ FRAMEWORKS 1996                 SUPPLEMENTARY
                                                                                                                   RESOURCES
4.1 All students will develop      4.1.A. Number Sense                   Pgs. 190-191, Overview
    number sense and will
    perform standard numerical     4.1.A.1 – Use real-life               Pg. 192, No. 11: Extend their
    operations and estimations     experiences, physical materials,      understanding of the number
    on all types of numbers in a   and technology to construct           system by constructing meanings
    variety of ways.               meanings for numbers (unless          for integers, rational numbers,
4.1.A. Number Sense                otherwise noted, all indicators       percents, exponents, roots,
4.1.B. Numerical Operations        for grade 6 pertain to these sets     absolute values, and numbers
4.1.C. Estimation                  of numbers as well).                  represented in scientific notation.
                                       • All integers
                                       • All fractions as part of a
                                          whole, as subset of a set,
                                          as a location on a number
                                          line, and as divisions of
                                          whole numbers
                                       • All decimals

                                   4.1.A.2 – Recognize the decimal       Pg. 192, No. 10: Understand
                                   nature of United States currency      money notations, count and
                                   and compute with money.               compute money, and recognize
                                                                         the decimal nature of United
                                                                         States currency.

                                   4.1.A.3 – Demonstrate a sense of      Pg. 193, No. 12: Develop
                                   the relative magnitudes of            number sense necessary for
                                   numbers.                              estimation.


                                                                       124
                                      Pg. 193, No. 13: Expand the
                                      sense of magnitudes of different
                                      number types to include integers,
                                      rational numbers, and roots.

4.1.A.4 – Explore the use of        Pg. 194, 14: Understand and
ratios and proportions in a variety apply ratios, proportions, and
of situations.                      percents in a variety of situations.
                                    Pg. 271, No. 11: Develop, apply
                                    and explain methods for solving
                                    problems involving proportions
                                    and percents.

4.1.A.5 – Understand and use          Pg. 192, No. 11: Extend their
whole-number percents between         understanding of the number
1 and 100 in a variety of             system by constructing meanings
situations.                           for integers, rational numbers,
                                      percents, exponents, roots,
                                      absolute values, and numbers
                                      represented in scientific notation.
                                      Pg. 194, 14: Understand and
                                      apply ratios, proportions, and
                                      percents in a variety of situations.
                                      Pg. 271, No. 11: Develop, apply
                                      and explain methods for solving
                                      problems involving proportions
                                      and percents.

4.1.A.6 – Use whole numbers,          Pg. 195, No. 18: Investigate the
fractions, and decimals to            relationships among fractions,
represent equivalent forms of the     decimals, and percents, and use
same number.                          all of them appropriately.
                                      Pg 324, No. 9: Use equivalent



                                    125
                                     representations of numbers such
                                     as fractions, decimals, and
                                     percents to facilitate estimation.

4.1.A.7 – Develop and apply          Pg. 195, No. 17: Develop and
number theory concepts in            apply number theory concepts,
problem solving situations.          such as, primes, factors, and
    • Primes, factors, multiples     multiples, in real-world and
    • Common multiples,              mathematical problem situations.
       common factors
    • Least common multiple,
       greatest common factor

4.1.A.8 – Compare and order          Pg. 194, No. 15: Develop and
numbers.                             use order relations for integers
                                     and rational numbers.


4.1.B. Numerical Operations          Pg. 268, Overview

4.1.B.1 – Recognize the              Pg. 269, No. 8: Extend their
appropriate use of each              understanding and use of
arithmetic operation in problem      arithmetic operations to fractions,
situations.                          decimals, integers, and rational
                                     numbers.

4.1.B.2 – Construct, use, and        Pg. 269, No. 6: Select and use
explain procedures for               appropriate computational
performing calculations with         methods from mental math,
fractions and decimals with:         estimation, paper-and-pencil, and
    • Pencil-and-paper               calculator methods, and check
    • Mental math                    the reasonableness of results.
    • Calculator


                                   126
4.1.B.3 – Use an efficient and       Pg. 269, No. 8: Extend their
accurate pencil-and-paper            understanding and use of
procedure for division of a 3-       arithmetic operations to fractions,
digit number by a 2-digit number.    decimals, integers, and rational
                                     numbers.


4.1.B.4 – Select pencil-and-         Pg. 269, No. 6: Select and use
paper, mental math, or a             appropriate computational
calculator as the appropriate        methods from mental math,
computational method in a given      estimation, paper-and-pencil, and
situation depending on the           calculator methods, and check
context and numbers.                 the reasonableness of results.


4.1.B.5 – Find squares and cubes     Pg. 270, No. 9: Extend their
of whole numbers.                    understanding of basic arithmetic
                                     operations on whole numbers to
                                     include powers and roots.


4.1.B.6 – Check the                  Pg. 269, No. 6: Select and use
reasonableness of results of         appropriate computational
computations.                        methods from mental math,
                                     estimation, paper-and-pencil, and
                                     calculator methods, and check
                                     the reasonableness of results.

4.1.B.7 – Understand and use the
various relationships among
operations and properties of
operations.



                                   127
4.1.B.8 – Understand and apply         Pg. 272, No. 12: Understand and
the standard algebraic order of        apply the standard algebraic
operations for the four basic          order of operations.
operations, including appropriate
use of parentheses.


4.1.C. Estimation                      Pg. 321, Overview

4.1.C.1 – Use a variety of             Pg. 323, No. 8: Develop, apply,
strategies for estimating both         and explain a variety of different
quantities and the results of          estimation strategies in problem
computations.                          situations involving quantities
                                       and measurement.


4.1.C.2 – Recognize when an            Pg. 322, No. 5: Recognize when
estimate is appropriate, and           estimation is appropriate, and
understand the usefulness of an        understand the usefulness of an
estimate as distinct from an exact     estimate as distinct from an exact
answer.                                answer.


4.1.C.3 – Determine the                Pg. 322, No. 6: Determine the
reasonableness of an answer by         reasonableness of an answer by
estimating the result of               estimating the result of
operations.                            operations.


4.1.C.4 – Determine whether a          Pg. 324, No. 10: Determine
given estimate is an overestimate      whether a given estimate is an
or an underestimate.                   overestimate or an underestimate.



                                     128
4.2 All students will develop       4.2.A. Geometric Properties             Pgs. 219-220, Overview
    spatial sense and the ability
    to use geometric properties,    4.2.A.1 – Understand and apply          Pg. 229, No. 14: Understand the
    relationships, and              concepts involving lines and            properties of lines and planes,
    measurement to model,           angles.                                 including parallel and
    describe and analyze                • Notation for line, ray,           perpendicular lines and planes,
    phenomena.                              angle, line segment             and intersecting lines and planes
4.2.A. Geometric Properties             • Properties of parallel,           and their angles of incidence.
4.2.B. Transforming Shapes                  perpendicular, and
4.2.C. Coordinate Geometry                  intersecting lines
4.2.D. Units of Measurement             • Sum of the measurements
4.2.E. Measuring Geometric                  of the interior angles of a
Objects                                     triangle is 180˚


                                    4.2.A.2 – Identify, describe,           Pg. 228, No. 13: Identify,
                                    compare, and classify polygons          describe, compare, and classify
                                    and circles.                            plane and solid geometric
                                        • Triangles by angles and           figures.
                                            sides
                                        • Quadrilaterals, including
                                            squares, rectangles,
                                            parallelograms,
                                            trapezoids, rhombi
                                        • Polygons by number of
                                            sides
                                        • Equilateral, equiangular,
                                            regular
                                        • All points equidistant
                                            from a given point form a
                                            circle




                                                                          129
4.2.A.3 – Identify similar figures.


4.2.A.4 – Understand and apply           Pg. 228, No. 12: Understand and
the concepts of congruence and           apply the concepts of symmetry,
symmetry (line and rotational).          similarity, and congruency.


4.2.A.5 – Compare properties of          Pg. 228, No. 13: Identify,
cylinders, prisms, cones,                describe, compare, and classify
pyramids, and spheres.                   plane and solid geometric
                                         figures.


4.2.A.6 – Identify, describe, and        Pg. 227, No. 11: Relate two-
draw the faces or shadows                dimensional and three-
(projections) of three-                  dimensional geometry using
dimensional geometric objects            shadows, perspectives,
from different perspectives.             projections, and maps.


4.2.A.7 – Identify a three-              Pg. 227, No. 11: Relate two-
dimensional shape with given             dimensional and three-
projections (top, front and side         dimensional geometry using
views).                                  shadows, perspectives,
                                         projections, and maps.

4.2.A.8 – Identify a three-
dimensional shape with a given
net (i.e., a flat pattern that folds
into a 3D shape).




                                       130
4.2.B. Transforming Shapes

4.2.B.1 – Use a translation, a           Pg. 229, No. 15: Explore the
reflection, or a rotation to map         relationships among geometric
one figure onto another                  transformations (translations,
congruent figure.                        reflections, rotations, and
                                         dilations), tessellations (tilings),
                                         and congruency and similarity.


4.2.B.2 – Recognize, identify,           Pg. 231, No. 19: Investigate,
and describe geometric                   explore, and describe the
relationships and properties as          geometry in nature and real-
they exist in nature, art, and other     world applications, using models,
real-world settings.                     manipulatives, and appropriate
                                         technology.

4.2.C. Coordinate Geometry

4.2.C.1 – Create geometric               Pg. 423, No. 9: Understand and
shapes with specified properties         use the rectangular coordinate
in the first quadrant on a               system.
coordinate grid.

4.2.D. Units of Measurement              Pgs. 294-295, Overview
                                         Pgs. 496-497, Overview

4.2.D.1 – Select and use                 Pg. 298, No. 14: Understand and
appropriate units to measure             apply measurement in their own
angles, area, surface area, and          lives and in other subject areas.
volume.                                  Pg. 230, No. 16: Develop,
                                         understand, and apply a variety
                                         of strategies for determining



                                       131
                                        perimeter, area, surface area,
                                        angle measure, and volume.

4.2.D.2 – Use a scale to find a         Pg. 296, No. 8: Read and
distance on a map or a length on        interpret various scales, including
a scale drawing.                        those based on number lines and
                                        maps.

4.2.D.3 – Convert measurement           Pg. 298, No. 13: Convert
units within a system (e.g., 3 feet     measurement units from one
= ____ inches).                         form to another, and carry out
                                        calculations that involve various
                                        units of measurement.

4.2.D.4 – Know approximate              Pg. 298, No. 13: Convert
equivalents between the standard        measurement units from one
and metric systems (e.g., one           form to another, and carry out
kilometer is approximately 6/10         calculations that involve various
of a mile).                             units of measurement.


4.2.D.5 – Use measurements and          Pg. 296, No. 7: Use estimated
estimates to describe and               and actual measurements to
compare phenomena.                      describe and compare
                                        phenomena.


4.2.E. Measuring Geometric              Pgs. 225-226, Overview
Objects

4.2.E.1 – Use a protractor to
measure angles.




                                      132
4.2.E.2 – Develop and apply            Pg. 230, No. 16: Develop,
strategies and formulas for            understand, and apply a variety
finding perimeter and area.            of strategies for determining
    • Triangle, square,                perimeter, area, surface area,
        rectangle, parallelogram,      angle measure, and volume.
        and trapezoid                  Pg. 297, No. 11: Develop
    • Circumference and area           formulas and procedures for
        of a circle                    solving problems related to
                                       measurement.


4.2.E.3 – Develop and apply            Pg. 230, No. 16: Develop,
strategies and formulas for            understand, and apply a variety
finding the surface area and           of strategies for determining
volume of rectangular prisms and       perimeter, area, surface area,
cylinders.                             angle measure, and volume.
                                       Pg. 297, No. 11: Develop
                                       formulas and procedures for
                                       solving problems related to
                                       measurement.


4.2.E.4 – Recognize that shapes        Pg. 299, No. 15: Understand and
with the same perimeter do not         explain the impact of the change
necessarily have the same area         of an object’s linear dimensions
and vice versa.                        on its perimeter, area, or volume.


4.2.E.5 – Develop informal ways        Pg. 297, No. 12: Explore
of approximating the measures of       situations involving quantities
familiar objects (e.g., use a grid     which cannot be measured
to approximate the area of the         directly or conveniently.
bottom of one’s foot).


                                     133
4.3 All students will represent      4.3.A. Patterns                      Pgs. 350-351, Overview
    and analyze relationships                                             Pgs. 418-419, Overview
    among variable quantities
    and solve problems               4.3.A.1 – Recognize, describe,       Pg. 195, No. 16: Recognize and
    involving patterns, functions,   extend, and create patterns          describe patterns in both finite
    and algebraic concepts and       involving whole numbers and          and infinite number sequences
    processes.                       rational numbers.                    involving whole numbers,
4.3.A. Patterns                          • Descriptions using tables,     rational numbers, and integers.
4.3.B. Functions and                         verbal rules, simple         Pg. 231, No. 18: Explore
Relationships                                equations, and graphs        patterns produced by processes of
4.3.C. Modeling                          • Formal iterative formulas      geometric change, relating
4.3.D. Procedures                            (e.g., NEXT = NOW * 3)       iteration, approximation, and
                                         • Recursive patterns,            fractals.
                                             including Pascal’s           Pg. 352, No. 7: Represent and
                                             Triangle (where each         describe mathematical
                                             entry is the sum of the      relationships with tables, rules,
                                             entries above it) and the    simple equations, and graphs.
                                             Fibonacci Sequence: 1, 1,    Pg. 356, No. 13: Develop,
                                             2, 3, 5, 8, … (where         analyze, and explain arithmetic
                                             NEXT = NOW +                 sequences.
                                             PREVIOUS)                    Pg. 466, No. 8: Experiment with
                                                                          iterative and recursive processes,
                                                                          with the aid of calculators and
                                                                          computers.
                                     4.3.B. Functions and
                                     Relationships

                                     4.3.B.1 – Describe the general       Pg. 355, No. 11: Understand and
                                     behavior of functions given by       describe the general behavior of
                                     formulas or verbal rules (e.g.,      functions.
                                     graph to determine whether
                                     increasing or decreasing, linear
                                     or not).


                                                                        134
4.3.C. Modeling

4.3.C.1 – Use patterns, relations,     Pg. 353, No. 8: Understand and
and linear functions to model          describe the relationships among
situations.                            various representations of
    • Using variables to               patterns and functions.
        represent unknown              Pg. 354, No. 9: Use patterns,
        quantities                     relationships, and functions to
    • Using concrete materials,        model situations and to solve
        tables, graphs, verbal         problems, in mathematics and in
        rules, algebraic               other subject areas.
        expressions/equations/in-      Pg. 355, No. 12: Use patterns,
        equalities                     relationships, and linear functions
                                       to model situations in
                                       mathematics and in other areas.


4.3.C.2 – Draw freehand sketches       Pg. 354, No. 10: Analyze
of graphs that model real              functional relationships to
phenomena and use such graphs          explain how a change in one
to predict and interpret events.       quantity results in a change in
    • Changes over time                another.
    • Relations between                Pg. 420, No. 6: Represent
        quantities                     situations and number patterns
    • Rates of change (e.g.,           with concrete materials, tables,
        when is plant growing          graphs, verbal rules, and standard
        slowly/rapidly, when is        algebraic notation.
        temperature dropping           Pg. 425, No. 13: Draw freehand
        most rapidly/slowly)           sketches of, and interpret, graphs
                                       which model real phenomena.




                                     135
                                   4.3.D. Procedures

                                   4.3.D.1 – Solve simple linear         Pg. 424, No. 10: Solve simple
                                   equations with manipulatives and      linear equations using concrete,
                                   informally.                           informal, and graphical methods,
                                       • Whole-number                    as well as appropriate paper-and-
                                          coefficients only, answers     pencil techniques.
                                          also whole numbers
                                       • Variables on one or both
                                          sides of equation

                                   4.3.D.2 – Understand and apply
                                   the properties of operations and
                                   numbers.
                                       • Distributive property
                                       • The product of a number
                                          and its reciprocal is 1


                                   4.3.D.3 – Evaluate numerical          Pg. 420, No. 5: Understand and
                                   expressions.                          use variables, expressions,
                                                                         equations, and inequalities.

                                   4.3.D.4 – Extend understanding        Pg. 420, No. 5: Understand and
                                   and use of inequality.                use variables, expressions,
                                       • Symbols (≥, ≠, ≤)               equations, and inequalities.
                                                                         Pg. 424, No. 12: Investigate
                                                                         inequalities and nonlinear
                                                                         equations informally.
4.4 All students will develop an   4.4.A. Data Analysis
    understanding of the
    concepts and techniques of     4.4.A.1 – Collect, generate,          Pg. 388, No. 9: Generate,
    data analysis, probability,    organize, and display data.           collect, organize, and analyze


                                                                       136
    and discrete mathematics,        •   Data generated from             data and represent this data in
    and will use them to model           surveys                         tables, charts, and graphs.
    situations, solve problems,
    and analyze and draw          4.4.A.2 – Read, interpret, select,     Pg. 388, No. 10: Select and use
    appropriate inferences from   construct, analyze, generate           appropriate graphical
    data.                         questions about, and draw              representations and measures of
4.4.A. Data Analysis              inferences from displays of data.      central tendency (mean, mode,
4.4.B. Probability                    • Bar graph, line graph,           and median) for sets of data.
4.4.C. Discrete Mathematics –            circle graph, table,            Pg. 389, No. 11: Make
Systematic Listing and Counting          histogram                       inferences and formulate and
4.4.D. Discrete Mathematics –         • Range, median, and mean          evaluate arguments based on data
Vertex-Edge Graphs and                • Calculators and                  analysis and data displays.
Algorithms                               computers used to record
                                         and process information

                                  4.4.A.3 – Respond to questions         Pg. 389, No. 11: Make
                                  about data, generate their own         inferences and formulate and
                                  questions and hypotheses, and          evaluate arguments based on data
                                  formulate strategies for               analysis and data displays.
                                  answering their questions and          Pg. 389, No. 12: Use lines of
                                  testing their hypotheses.              best fit to interpolate and predict
                                                                         from data.

                                  4.4.B. Probability                     Pgs. 386-387, Overview

                                  4.4.B.1 – Determine probabilities      Pg. 391, No. 16: Interpret
                                  of events.                             probabilities as ratios and
                                      • Event, complementary             percents.
                                         event, probability of an
                                         event
                                      • Multiplication rule for
                                         probabilities
                                      • Probability of certain


                                                                       137
       event is 1 and of
       impossible event is 0

   •   Probability of event and
       complementary event add
       up to 1

4.4.B.2 – Determine probability        Pg. 390, No. 15: Use models of
using intuitive, experimental, and     probability to predict events
theoretical methods (e.g., using       based on actual data.
model of picking items of              Pg. 391, No. 16: Interpret
different colors from a bag).          probabilities as ratios and
    • Given numbers of various         percents.
        types of items in a bag,
        what is the probability
        that an item of one type
        will be picked
    • Given data obtained
        experimentally, what is
        the likely distribution of
        items in the bag

4.4.B.3 – Explore compound             Pg. 389, No. 13: Determine the
events.                                probability of a compound event.

4.4.B.4 – Model situations             Pg. 390, No. 14: Model
involving probability using            situations involving probability,
simulations (with spinners, dice)      such as genetics, using both
and theoretical models.                simulations and theoretical
                                       models.
4.4.B.5 – Recognize and
understand the connections
among the concepts of


                                     138
independent outcomes, picking at
random, and fairness.

4.4.C. Discrete Mathematics –         Pgs. 462-463, Overview
Systematic Listing and
Counting

4.4.C.1 – Solve counting              Pg. 464, No. 6: Use systematic
problems and justify that all         listing, counting, and reasoning
possibilities have been               in a variety of different contexts.
enumerated without duplication.
    • Organized lists, charts,
        three diagrams, tables
    • Venn diagrams


4.4.C.2 – Apply the
multiplication principle of
counting.
    • Simple situations (e.g.,
       you can make 3 X 4 = 12
       outfits using 3 shirts and
       4 skirts)
    • Number of ways a
       specified number of items
       can be arranged in order
       (concept of permutation)
    • Number of ways of
       selecting a slate of
       officers from a class (e.g.,
       if there are 23 students
       and 3 officers, the number
       is 23 X 22 X 21)


                                   139
4.4.C.3 – List the possible
combinations of two elements
chosen from a given set (e.g.,
forming a committee of two from
a group of 12 students, finding
how many handshakes there will
be among ten people if everyone
shakes each other person’s hand
once).

4.4.D. Discrete Mathematics –
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Devise strategies for         Pg. 465, No. 7: Recognize
winning simple games (e.g., start       common discrete mathematical
with two piles of objects, each of      models, explore their properties,
two players in turn removes any         and design them for specific
number of objects from a single         situations.
pile, and the person to take the
last group of objects wins) and
express those strategies as sets of
directions.

4.4.D.2 – Analyze vertex-edge
graphs and tree diagrams.
    • Can a picture or a vertex-
       edge graph be drawn with
       a single line? (degree of
       vertex)
    • Can you get from any
       vertex to any other
       vertex? (connectedness)


                                      140
4.4.D.3 – Use vertex-edge graphs
to find solutions to practical
problems.
    • Delivery route that stops
        at specified sites but
        involves least travel
    • Shortest route from one
        site on a map to another




                                   141
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                     142
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                     143
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             144
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       145
                                                              MATHEMATICS
                                                                GRADE 7


         STANDARD                       STUDENT OUTCOME                       SUGGESTED ACTIVITIES               TEACHER’S NOTES AND
                                                                               NJ FRAMEWORKS 1996                  SUPPLEMENTARY
                                                                                                                     RESOURCES
4.1 All students will develop       4.1.A. Number Sense                   Pgs. 197-198, Overview
    number sense and will
    perform standard numerical      4.1.A.1 – Extend understanding        Pg. 199, No. 11: Extend their
    operations and estimations on   of the number system by               understanding of the number
    all types of numbers in a       constructing meanings for the         system by constructing meanings
    variety of ways.                following (unless otherwise           for integers, rational numbers,
4.1.A. Number Sense                 noted, all indicators for grade 7     percents, exponents, roots,
4.1.B. Numerical Operations         pertain to these sets of numbers      absolute values, and numbers
4.1.C. Estimation                   as well).                             represented in scientific notation.
                                        • Rational numbers                Pg. 201, No. 15: Develop and
                                        • Percents                        use order relations for integers
                                        • Whole numbers with              and rational numbers.
                                            exponents

                                    4.1.A.2 – Demonstrate a sense of      Pg. 200, No. 13: Expand the
                                    the relative magnitudes of            sense of magnitudes of different
                                    numbers.                              number types to include integers,
                                                                          rational numbers, and roots.

                                    4.1.A.3 – Understand and use          Pg. 201, No. 14: Understand and
                                    ratios, proportions, and percents     apply ratios, proportions, and
                                    (including percents greater than      percents in a variety of situations.
                                    100 and less than 1) in a variety     Pg. 275, No. 11: Develop, apply
                                    of situations.                        and explain the methods for
                                                                          solving problems involving
                                                                          proportions and percents.


                                                                        146
                                      Pg. 303, No. 12: Explore
                                      situations involving quantities
                                      which cannot be measured
                                      directly or conveniently.

4.1.A.4 – Compare and order           Pg. 203, No. 19: Identify, derive,
numbers of all named types.           and compare properties of
                                      numbers.

4.1.A.5 – Use whole numbers,          Pg. 199, No. 10: Understand
fractions, decimals, and percents     money notations, count and
to represent equivalent forms of      compute money, and recognize
the same number.                      the decimal nature of United
                                      States currency.
                                      Pg. 203, No. 18: Investigate the
                                      relationships among fractions,
                                      decimals, and percents, and use
                                      all of them appropriately.
                                      Pg. 274, No. 8: Extend their
                                      understanding and use of
                                      arithmetic operations to fractions,
                                      decimals, integers, and rational
                                      numbers.
                                      Pg. 328, No. 9: Use equivalent
                                      representations of numbers such
                                      as fractions, decimals, and
                                      percents to facilitate estimation.


4.1.A.6 – Understand that all         Pg. 202, No. 17: Develop and
fractions can be represented as       apply number theory concepts,
repeating or terminating              such as primes, factors, and
decimals.                             multiples, in real-world and



                                    147
                                       mathematical problem situations.
                                       Pg. 505, No. 6: Investigate,
                                       represent, and use non-
                                       terminating decimals.

4.1.B. Numerical Operations            Pg. 273, Overview

4.1.B.1 – Use and explain              Pg. 274, No. 6: Select and use
procedures for performing              appropriate computational
calculations with integers and all     methods from mental math,
number types named above with:         estimation, paper-and-pencil, and
    • Pencil-and-paper                 calculator methods, and check
    • Mental math                      the reasonableness of results.
    • Calculator


4.1.B.2 – Use exponentiation to        Pg. 274, No. 9: Extend their
find whole number powers of            understanding of basic arithmetic
numbers.                               operations on whole numbers to
                                       include powers and roots.

4.1.B.3 – Understand and apply
the standard algebraic order of
operations, including appropriate
use of parentheses.


4.1.C. Estimation                      Pg. 325, Overview

4.1.C.1 – Use equivalent               Pg. 200, No. 12: Develop
representations of numbers such        number sense necessary for
as fractions, decimals, and            estimation.
percents to facilitate estimation.


                                     148
                                                                           Pg. 275, No. 10: Develop, apply,
                                                                           and explain procedures for
                                                                           computation and estimation with
                                                                           whole numbers, fractions,
                                                                           decimals, integers, and rational
                                                                           numbers.

4.2 All students will develop          4.2.A. Geometric Properties         Pgs. 233-234, Overview
    spatial sense and the ability to
    use geometric properties,          4.2.A.1 – Understand and apply
    relationships, and                 properties of polygons.
    measurement to model,                  • Quadrilaterals, including
    describe and analyze                      squares, rectangles,
    phenomena.                                parallelograms,
4.2.A. Geometric Properties                   trapezoids, rhombi
4.2.B. Transforming Shapes                 • Regular polygons
4.2.C. Coordinate Geometry
4.2.D. Units of Measurement            4.2.A.2 – Understand and apply      Pg. 235, No. 12: Understand and
4.2.E. Measuring Geometric             the concept of similarity.          apply the concepts of symmetry,
Objects                                    • Using proportions to find     similarity, and congruence.
                                              missing measures             Pg. 302, No. 8: Read and
                                           • Scale drawings                interpret various scales, including
                                           • Models of 3D objects          those based on number lines and
                                                                           maps.
                                                                           Pg. 235, No. 11: Relate two-
                                                                           dimensional and three-
                                                                           dimensional geometry using
                                                                           shadows, perspectives,
                                                                           projections, and maps.
                                                                           Pg. 236, No. 13: Identify,
                                                                           describe, compare, and classify
                                                                           plane and solid geometric
                                                                           figures.


                                                                         149
                                      Pg. 305, No. 16: Apply their
                                      knowledge of measurement to the
                                      construction of a variety of two-
                                      and three-dimensional figures.


4.2.A.3 – Use logic and
reasoning to make and support
conjectures about geometric
objects.


4.2.B. Transforming Shapes

4.2.B.1 – Understand and apply        Pg. 237, No. 15: Explore the
transformations.                      relationships among geometric
    • Finding the image, given        transformations (translations,
        the pre-image, and vice       reflections, rotations, and
        versa                         dilations), tessellations (tilings),
    • Sequence of                     and congruence and similarity.
        transformations needed to
        map one figure onto
        another
    • Reflections, rotations, and
        translations result in
        images congruent to the
        pre-image
    • Dilations
        (stretching/shrinking)
        result in images similar to
        the pre-image




                                  150
4.2.C. Coordinate Geometry

4.2.C.1 – Use coordinates in four     Pg. 431, No. 9: Understand and
quadrants to represent geometric      use the rectangular coordinate
concepts.                             system.

4.2.C.2 – Use a coordinate grid to
model and quantify
transformations (e.g., translate
right 4 units).

4.2.D. Units of Measurement           Pg. 301, Overview

4.2.D.1 – Solve problems
requiring calculations that
involve different units of
measurement within a
measurement system (e.g., 4’3”
plus 7’10” equals 12”1”).

4.2.D.2 – Select and use              Pg. 302, No. 9: Determine the
appropriate units and tools to        degree of accuracy needed in a
measure quantities to the degree      given situation and choose units
of precision needed in a              accordingly.
particular problem-solving
situation.

4.2.D.3 – Recognize that all          Pg. 302, No. 10: Understand that
measurements of continuous            all measurements of continuous
quantities are approximations.        quantities are approximate.
                                      Pg. 506, No. 8: Approximate
                                      quantities with increasing
                                      degrees of accuracy.



                                    151
                                    4.2.E. Measuring Geometric
                                    Objects

                                    4.2.E.1 – Develop and apply            Pg. 238, No. 16: Develop,
                                    strategies for finding perimeter       understand, and apply a variety
                                    and area.                              of strategies for determining
                                        • Geometric figures made           perimeter, area, surface area,
                                            by combining triangles,        angle measure, and volume.
                                            rectangles and circles or      Pg. 302, No. 7: Use estimated
                                            parts of circles               and actual measurements to
                                        • Estimation of area using         describe and compare
                                            grids of various sizes         phenomena.

                                    4.2.E 2 – Recognize that the
                                    volume of a pyramid or cone is
                                    one-third of the volume of the
                                    prism or cylinder with the same
                                    base and height (e.g., use rice to
                                    compare volumes of figures with
                                    the same base and height.
4.3 All students will represent     4.3.A. Patterns                        Pg. 357, Overview
    and analyze relationships
    among variable quantities and   4.3.A.1 – Recognize, describe,         Pg. 358, No. 8: Understand and
    solve problems involving        extend, and create patterns            describe the relationships among
    patterns, functions, and        involving whole numbers,               various representations of
    algebraic concepts and          rational numbers, and integers.        patterns and functions.
    processes.                          • Descriptions using tables,       Pg. 202, No. 16: Recognize and
4.3.A. Patterns                             verbal and symbolic rules,     describe patterns in both finite
4.3.B. Functions and                        graphs, simple equations       and infinite number sequences
Relationships                               or expressions                 involving whole numbers,
4.3.C. Modeling                         • Finite and infinite              rational numbers, and integers.
4.3.D. Procedures                           sequences


                                                                         152
   •   Generating sequences by        Pg. 504, No. 5: Develop an
       using calculators to           understanding of infinite
       repeatedly apply a             sequences that arise in natural
       formula                        situations.
                                      Pg. 362, No. 13: Develop,
                                      analyze, and explain arithmetic
                                      sequences.


4.3.B. Functions and                  Pgs. 427-428, Overview
Relationships

4.3.B.1 – Graph functions, and        Pg. 360, No. 11: Understand and
understand and describe their         describe the general behavior of
general behavior.                     functions.
    • Equations involving two         Pg. 430, No. 8: Analyze tables
       variables                      and graphs to identify properties
                                      and relationships.


4.3.C. Modeling

4.3.C.1 – Analyze functional          Pg. 360, No. 10: Analyze
relationships to explain how a        functional relationships to
change in one quantity can result     explain how a change in one
in a change in another, using         quantity results in a change in
pictures, graphs, charts, and         another.
equations.                            Pg. 505, No. 7: Represent,
                                      analyze, and predict relations
                                      between quantities, especially
                                      quantities changing over time.




                                    153
4.3.C.2 – Use patterns, relations,     Pg. 304, No. 15: Understand and
symbolic algebra, and linear           explain the impact of the change
functions to model situations.         of an object’s linear dimensions
    • Using manipulatives,             on its perimeter, area, or volume.
       tables, graphs, verbal          Pg. 358, No. 7: Represent and
       rules, algebraic                describe mathematical
       expressions/equations/in-       relationships with tables, rules,
       equalities                      simple equations, and graphs.
    • Growth situations, such as       Pg. 361, No. 12: Use patterns,
       population growth and           relationships, and linear functions
       compound interest, using        to model situations in
       recursive (e.g., NOW-           mathematics and in other areas.
       NEXT) formulas (cf.             Pg. 429, No. 5: Understand and
       science standard 5.5 and        use variables, expressions,
       social studies standard         equations, and inequalities.
       6.6)                            Pg. 429, No. 6: Represent
                                       situations and number patterns
                                       with concrete materials, tables,
                                       graphs, verbal rules, and standard
                                       algebraic notation.
                                       Pg. 504, No. 4: Recognize and
                                       express the difference between
                                       linear and exponential growth.

4.3.D. Procedures

4.3.D.1 – Use graphing                 Pg. 430, No. 7: Use graphing
techniques on a number line.           techniques on a number line to
    • Absolute value                   model both absolute value and
    • Arithmetic operations            arithmetic operations.
       represented by vectors          Pg. 432, No. 12: Investigate
       (arrows) (e.g., “-3 + 6” is     inequalities and nonlinear
       “left 3, right 6”)              equations informally.



                                     154
                                   4.3.D.2 – Solve simple linear         Pg. 432, No. 10: Solve simple
                                   equations informally and              linear equations using concrete,
                                   graphically.                          informal, and graphical methods,
                                       • Multi-step, integer             as well as appropriate paper-and-
                                          coefficients only              pencil techniques.
                                          (although answers may          Pg. 432, No. 11: Explore linear
                                          not be integers)               equations through the use of
                                       • Using paper-and-pencil,         calculators, computers, and other
                                          calculators, graphing          technology.
                                          calculators, spreadsheets,
                                          and other technology

                                   4.3.D.3 – Create, evaluate, and
                                   simplify algebraic expressions
                                   involving variables.
                                       • Order of operations,
                                          including appropriate use
                                          of parentheses
                                       • Substitution of a number
                                          for a variable

                                   4.3.D.4 – Understand and apply        Pg. 203, No. 19: Identify, derive,
                                   the properties of operations,         and compare properties of
                                   numbers, equations, and               numbers.
                                   inequalities                          Pg. 432, No. 12: Investigate
                                       • Additive inverse                inequalities and nonlinear
                                       • Multiplicative inverse          equations informally.

4.4 All students will develop an  4.4.A. Data Analysis
    understanding of the concepts
    and techniques of data        4.4.A.1 – Select and use               Pg. 394, No. 9: Generate,
    analysis, probability, and    appropriate representations for        collect, organize, and analyze


                                                                       155
    discrete mathematics, and     sets of data, and measures of          data and represent this data in
    will use them to model        central tendency (mean, median,        tables, charts, and graphs.
    situations, solve problems,   and mode).                             Pg. 394, No. 10: Select and use
    and analyze and draw              • Type of display most             appropriate graphical
    appropriate inferences from           appropriate for given data     representations and measures of
    data.                             • Box-and-whisker plot,            central tendency (mean, mode,
4.4.A. Data Analysis                      upper quartile, lower          and median) for sets of data.
4.4.B. Probability                        quartile
4.4.C. Discrete Mathematics –         • Scatter plot
Systematic Listing and Counting       • Calculators and computer
4.4.D. Discrete Mathematics –             used to record and
Vertex-Edge Graphs and                    process information
Algorithms

                                  4.4.A.2 – Make inferences and          Pg. 395, No. 11: Make
                                  formulate and evaluate                 inferences and formulate and
                                  arguments based on displays and        evaluate arguments based on data
                                  analysis of data.                      analysis and data displays.

                                  4.4.B. Probability                     Pgs. 392-393, Overview

                                  4.4.B.1 – Interpret probabilities      Pg. 396, No. 16: Interpret
                                  as ratios, percents, and decimals.     probabilities as ratios and
                                                                         percents.

                                  4.4.B.2 – Model situations             Pg. 396, No. 15: Use models of
                                  involving probability with             probability to predict events
                                  simulations (using spinners, dice,     based on actual data.
                                  calculators and computers) and
                                  theoretical models.
                                      • Frequency, relative
                                          frequency



                                                                       156
4.4.B.3 – Estimate probabilities       Pg. 396, No. 14: Model
and make predictions based on          situations involving probability,
experimental and theoretical           such as genetics, using both
probabilities.                         simulations and theoretical
                                       methods.

4.4.B.4 – Play and analyze
probability-based games, and
discuss the concepts of fairness
and expected value.

4.4.C. Discrete Mathematics –          Pgs. 471-472, Overview
Systematic Listing and
Counting

4.4.C.1 – Apply the                    Pg. 473, No. 6: Use systematic
multiplication principle of            listing, counting, and reasoning
counting.                              in a variety of different contexts.
    • Permutations: ordered
       situations with
       replacement (e.g., number
       of possible license plates)
       vs. ordered situations
       without replacement (e.g.,
       number of possible slates
       of 3 class officers from a
       23-student class)

4.4.C.2 – Explore counting             Pg. 476, No. 9: Explore methods
problems involving Venn                for storing, processing, and
diagrams with three attributes         communicating information.
(e.g., there are 15, 20, and 25
students respectively in the chess


                                     157
club, the debating team, and the
engineering society; how many
different students belong to the
three clubs if there are 6 students
in chess and debating, 7 students
in chess and engineering, 8
students in debating and
engineering, and 2 students in all
three?).

4.4.C.3 – Apply techniques of           Pg. 359, No. 9: Use patterns,
systematic listing, counting, and       relationships, and functions to
reasoning in a variety of different     model situations and to solve
contexts.                               problems in mathematics and in
                                        other subject areas.

4.4.D. Discrete Mathematics –
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Use vertex-edge graphs        Pg. 474, No. 7: Recognize
to represent and find solutions to      common discrete mathematical
practical problems.                     models, explore their properties,
    • Finding the shortest              and design them for specific
        network connecting              situations.
        specified sites                 Pg. 477, No. 10: Devise,
    • Finding the shortest route        describe, and test algorithms for
        on a map from one site to       solving optimization and search
        another                         problems.
    • Finding the shortest
        circuit on a map that
        makes a tour of specified
        sites


                                      158
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                     159
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




                                     160
                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             161
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       162
                                                              MATHEMATICS
                                                                GRADE 8


         STANDARD                       STUDENT OUTCOME                      SUGGESTED ACTIVITIES               TEACHER’S NOTES AND
                                                                              NJ FRAMEWORKS 1996                  SUPPLEMENTARY
                                                                                                                    RESOURCES
4.1 All students will develop       4.1.A. Number Sense                  Pgs. 197-198, Overview
    number sense and will
    perform standard numerical      4.1.A.1 – Extend understanding       Pg. 199, No. 11: Extend their
    operations and estimations on   of the number system by              understanding of the number
    all types of numbers in a       constructing meanings for the        system by constructing meanings
    variety of ways.                following (unless otherwise          for integers, rational numbers,
4.1.A. Number Sense                 noted, all indicators for grade 8    percents, exponents, roots,
4.1.B. Numerical Operations         pertain to these sets of numbers     absolute values, and numbers
4.1.C. Estimation                   as well).                            represented in scientific notation.
                                        • Rational numbers
                                        • Percents
                                        • Exponents
                                        • Roots
                                        • Absolute values
                                        • Numbers represented in
                                            scientific notation

                                    4.1.A.2 – Demonstrate a sense of     Pg. 200, No. 13: Expand the
                                    the relative magnitudes of           sense of magnitudes of different
                                    numbers.                             number types to include integers,
                                                                         rational numbers, and roots.


                                    4.1.A.3 – Understand and use         Pg. 201, No. 14: Understand and
                                    ratios, rates, proportions, and      apply ratios, proportions, and
                                    percents (including percents         percents in a variety of situations.


                                                                       163
greater than 100 and less than 1)     Pg. 275, No. 11: Develop, apply
in a variety of situations.           and explain the methods for
                                      solving problems involving
                                      proportions and percents.
                                      Pg. 303, No. 12: Explore
                                      situations involving quantities
                                      which cannot be measured
                                      directly or conveniently.

4.1.A.4 – Compare and order           Pg. 203, No. 19: Identify, derive,
numbers of all named types.           and compare properties of
                                      numbers.


4.1.A.5 – Use whole numbers,          Pg. 199, No. 10: Understand
fractions, decimals, and percents     money notations, count and
to represent equivalent forms of      compute money, and recognize
the same number.                      the decimal nature of United
                                      States currency.
                                      Pg. 203, No. 18: Investigate the
                                      relationships among fractions,
                                      decimals, and percents, and use
                                      all of them appropriately.
                                      Pg. 274, No. 8: Extend their
                                      understanding and use of
                                      arithmetic operations to fractions,
                                      decimals, integers, and rational
                                      numbers.
                                      Pg. 328, No. 9: Use equivalent
                                      representations of numbers such
                                      as fractions, decimals, and
                                      percents to facilitate estimation.




                                    164
4.1.A.6 – Recognize that
repeating decimals correspond to
fractions and determine their
fractional equivalents.
    • 5/7 = 0.714285714285…
       = 0.
                                        Pg. 505, No. 6: Investigate,
4.1.A.7 – Construct meanings for
                                        represent, and use non-
common irrational numbers, such
                                        terminating decimals.
as π (pi) and the square root of 2.
                                        Pg. 273, Overview
4.1.B. Numerical Operations
                                        Pg. 274, No. 6: Select and use
4.1.B.1 – Use and explain
                                        appropriate computational
procedures for performing
                                        methods from mental math,
calculations involving addition,
                                        estimation, paper-and-pencil, and
subtraction, multiplication,
                                        calculator methods, and check
division, and exponentiation with
                                        the reasonableness of results.
integers and all number types
named above with:
    • Pencil-and-paper
    • Mental math
    • Calculator
                                        Pg. 274, No. 9: Extend their
4.1.B.2 – Use exponentiation to         understanding of basic arithmetic
find whole number powers of             operations on whole numbers to
numbers.                                include powers and roots.


4.1.B.3 – Find square and cube
roots of numbers and understand
the inverse nature of powers and
roots.


                                      165
4.1.B.4 – Solve problems               Pg. 275, No. 11: Develop, apply,
involving proportions and              and explain methods for solving
percents.                              problems involving proportions
                                       and percents.


4.1.B.5 – Understand and apply         Pg. 276, No. 12: Understand and
the standard algebraic order of        apply the standard algebraic
operations, including appropriate      order of operations.
use of parentheses.

4.1.C. Estimation                      Pg. 325, Overview

4.1.C.1 – Estimate square and
cube roots of numbers.


4.1.C.2 – Use equivalent               Pg. 200, No. 12: Develop
representations of numbers such        number sense necessary for
as fractions, decimals, and            estimation.
percents to facilitate estimation.     Pg. 275, No. 10: Develop, apply,
                                       and explain procedures for
                                       computation and estimation with
                                       whole numbers, fractions,
                                       decimals, integers, and rational
                                       numbers.


4.1.C.3 – Recognize the                Pg. 326, No. 5: Recognize when
limitations of estimation and          estimation is appropriate, and
assess the amount of error             understand the usefulness of an
resulting from estimation.             estimate as distinct from an exact
                                       answer.



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                                                                             Pg. 326, No. 6: Determine the
                                                                             reasonableness of an answer by
                                                                             estimating the result of
                                                                             operations.
                                                                             Pg. 328, No. 10: Determine
                                                                             whether a given estimate is an
                                                                             overestimate or an underestimate.
4.2 All students will develop          4.2.A. Geometric Properties           Pgs. 233-234, Overview
    spatial sense and the ability to
    use geometric properties,          4.2.A.1 – Understand and apply        Pg. 237, No. 14: Understand the
    relationships, and                 concepts involving lines, angles,     properties of lines and planes,
    measurement to model,              and planes.                           including parallel and
    describe and analyze                   • Complementary and               perpendicular lines and planes,
    phenomena.                                supplementary angles           and intersecting lines and planes
4.2.A. Geometric Properties                • Vertical angles                 and their angles of incidence.
4.2.B. Transforming Shapes                 • Bisectors and
4.2.C. Coordinate Geometry                    perpendicular bisectors
4.2.D. Units of Measurement                • Parallel, perpendicular,
4.2.E. Measuring Geometric                    and intersecting planes
Objects                                    • Intersection of plane with
                                              cube, cylinder, cone, and
                                              sphere

                                       4.2.A.2 – Understand and apply        Pg. 238, No. 17: Understand and
                                       the Pythagorean Theorem.              apply the Pythagorean Theorem.


                                       4.2.A.3 – Understand and apply
                                       properties of polygons.
                                           • Quadrilaterals, including
                                              squares, rectangles,
                                              parallelograms,
                                              trapezoids, rhombi


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   •   Regular polygons
   •   Sum of measures of
       interior angles of a
       polygon
   •   Which polygons can be
       used alone to generate a
       tessellation and why


4.2.A.4 – Understand and apply       Pg. 303, No. 12: Explore
the concept of similarity.           situations involving quantities
     • Using proportions to find     which cannot be measured
        missing measures             directly or conveniently.
     • Scale drawings                Pg. 302, No. 8: Read and
     • Models of 3D objects          interpret various scales, including
                                     those based on number lines and
                                     maps.
                                     Pg.
                                     Pg. 235, No. 11: Relate two-
                                     dimensional and three-
                                     dimensional geometry using
                                     shadows, perspectives,
                                     projections, and maps.
                                     Pg. 236, No. 13: Identify,
                                     describe, compare, and classify
                                     plane and solid geometric
                                     figures.
                                     Pg. 305, No. 16: Apply their
                                     knowledge of measurement to the
                                     construction of a variety of two-
                                     and three-dimensional figures.




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4.2.A.5 – Use logic and
reasoning to make and support
conjectures about geometric
objects.

4.2.A.6 – Perform basic
geometric constructions using a
variety of methods (e.g.,
straightedge and compass,
patty/tracing paper, or
technology).
    • Congruent angles or line
        segments
    • Midpoint of a line
        segment

4.2.A.7 – Create two-
dimensional representations (e.g.,
nets or projective views) for the
surfaces of three-dimensional
objects.


4.2.B. Transforming Shapes

4.2.B.1 – Understand and apply        Pg. 237, No. 15: Explore the
transformations.                      relationships among geometric
    • Finding the image, given        transformations (translations,
        the pre-image, and vice       reflections, rotations, and
        versa                         dilations), tessellations (tilings),
    • Sequence of                     and congruence and similarity.
        transformations needed to
        map one figure onto


                                    169
       another
   •   Reflections, rotations, and
       translations result in
       images congruent to the
       pre-image
   •   Dilations
       (stretching/shrinking)
       result in images similar to
       the pre-image

4.2.B.2 – Use iterative               Pg. 238, No. 18: Explore
procedures to generate geometric      patterns produced by processes of
patterns.                             geometric change, relating
    • Fractals (e.g., the Koch        iteration, approximation, and
        Snowflake)                    fractals.
    • Self-similarity
    • Construction of initial
        stages

   •   Patterns in successive
       stages (e.g., number of
       triangles in each stage of
       Sierpinski’s Triangle)


4.2.C. Coordinate Geometry

4.2.C.1 – Use coordinates in four     Pg. 431, No. 9: Understand and
quadrants to represent geometric      use the rectangular coordinate
concepts.                             system.

4.2.C.2 – Use a coordinate grid to
model and quantify


                                    170
transformations (e.g., translate
right 4 units).

4.2.D. Units of Measurement          Pg. 301, Overview

4.2.D.1 – Solve problems             Pg. 303, No. 13: Convert
requiring calculations that          measurement units from one
involve different units of           form to another, and carry out
measurement within a                 calculations that involve various
measurement system (e.g., 4’3”       units of measurement.
plus 7’10” equals 12”1”).

4.2.D.2 – Use approximate
equivalents between standard and
metric systems to estimate
measurements (e.g., 5 kilometers
is about 3 miles).

4.2.D.3 – Recognize that the         Pg. 506, No. 8: Approximate
degree of precision needed in        quantities with increasing
calculations depends on how the      degrees of accuracy.
results will be used and the
instruments used to generate the
measurements.


4.2.D.4 – Select and use             Pg. 302, No. 9: Determine the
appropriate units and tools to       degree of accuracy needed in a
measure quantities to the degree     given situation and choose units
of precision needed in a             accordingly.
particular problem-solving
situation.




                                   171
4.2.D.5 – Recognize that all          Pg. 302, No. 10: Understand that
measurements of continuous            all measurements of continuous
quantities are approximations.        quantities are approximate.


4.2.D.6 – Solve problems that
involve compound measurement
units, such as speed (miles per
hour), air pressure (pounds per
square inch), and population
density (persons per square mile).

4.2.E. Measuring Geometric
Objects

4.2.E.1 – Develop and apply           Pg. 238, No. 16: Develop,
strategies for finding perimeter      understand, and apply a variety
and area.                             of strategies for determining
    • Geometric figures made          perimeter, area, surface area,
        by combining triangles,       angle measure, and volume.
        rectangles and circles or     Pg. 302, No. 7: Use estimated
        parts of circles              and actual measurements to
    • Estimation of area using        describe and compare
        grids of various sizes        phenomena.
    • Impact of a dilation on
        the perimeter and area of
        a 2-dimensional figure


4.2.E.2 – Recognize that the
volume of a pyramid or cone is
one-third of the volume of the


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                                    prism or cylinder with the same
                                    base and height (e.g., use rice to
                                    compare volumes of figures with
                                    the same base and height.


                                    4.2.E.3 – Develop and apply            Pg. 303, No. 11: Develop
                                    strategies and formulas for            formulas and procedures for
                                    finding the surface area and           solving problems related to
                                    volume of a three-dimensional          measurement.
                                    figure.                                Pg. 507, No. 10: Develop
                                        • Volume – prism, cone,            informal ways of approximating
                                            pyramid                        the surface area and volume of
                                        • Surface area – prism             familiar objects, and discuss
                                            (triangular or rectangular     whether the approximations
                                            base), pyramid (triangular     make sense.
                                            or rectangular base)

                                       •   Impact of a dilation on         Pg. 508, No. 11: Express
                                           the surface area and            mathematically and explain the
                                           volume of a three-              impact of the change of an
                                           dimensional figure              object’s linear dimensions on its
                                                                           surface area and volume.

                                    4.2.E.4 – Use formulas to find
                                    the volume and surface area of a
                                    sphere.
4.3 All students will represent     4.3.A. Patterns                        Pg. 357, Overview
    and analyze relationships
    among variable quantities and   4.3.A.1 – Recognize, describe,         Pg. 358, No. 8: Understand and
    solve problems involving        extend, and create patterns            describe the relationships among
    patterns, functions, and        involving whole numbers,               various representations of
    algebraic concepts and          rational numbers, and integers.        patterns and functions.


                                                                         173
    processes.            •   Descriptions using tables,    Pg. 202, No. 16: Recognize and
4.3.A. Patterns               verbal and symbolic rules,    describe patterns in both finite
4.3.B. Functions and          graphs, simple equations      and infinite number sequences
Relationships                 or expressions                involving whole numbers,
4.3.C. Modeling           •   Finite and infinite           rational numbers, and integers.
4.3.D. Procedures             sequences                     Pg. 504, No. 5: Develop an
                          •   Arithmetic sequences          understanding of infinite
                              (i.e., sequences generated    sequences that arise in natural
                              by repeated addition of a     situations.
                              fixed number, positive or     Pg. 362, No. 13: Develop,
                              negative)                     analyze, and explain arithmetic
                          •   Geometric sequences           sequences.
                              (i.e., sequences generated
                              by repeated multiplication
                              by a fixed positive ratio,
                              greater than 1 or less than
                              1)

                          •   Generating sequences by
                              using calculators to
                              repeatedly apply a
                              formula

                       4.3.B. Functions and                 Pgs. 427-428, Overview
                       Relationships

                       4.3.B.1 – Graph functions, and       Pg. 360, No. 11: Understand and
                       understand and describe their        describe the general behavior of
                       general behavior.                    functions.
                           • Equations involving two        Pg. 430, No. 8: Analyze tables
                              variables                     and graphs to identify properties
                                                            and relationships.
                           • Rates of change (informal
                              notion of slope)


                                                         174
4.3.B.2 – Recognize and describe      Pg. 430, No. 8: Analyze tables
the difference between linear and     and graphs to identify properties
exponential growth, using tables,     and relationships.
graphs, and equations.

4.3.C. Modeling

4.3.C.1 – Analyze functional          Pg. 304, No. 15: Understand and
relationships to explain how a        explain the impact of the change
change in one quantity can result     of an object’s linear dimensions
in a change in another, using         on its perimeter, area, or volume.
pictures, graphs, charts, and         Pg. 360, No. 10: Analyze
equations.                            functional relationships to
                                      explain how a change in one
                                      quantity results in a change in
                                      another.

4.3.C.2 – Use patterns, relations,    Pg. 358, No. 7: Represent and
symbolic algebra, and linear          describe mathematical
functions to model situations.        relationships with tables, rules,
    • Using concrete materials        simple equations, and graphs.
       (manipulatives), tables,       Pg. 361, No. 12: Use patterns,
       graphs, verbal rules,          relationships, and linear functions
       algebraic                      to model situations in
       expressions/equations/in-      mathematics and in other areas.
       equalities                     Pg. 429, No. 6: Represent
    • Growth situations, such as      situations and number patterns
       population growth and          with concrete materials, tables,
       compound interest, using       graphs, verbal rules, and standard
       recursive (e.g., NOW-          algebraic notation.
       NEXT) formulas (cf.            Pg. 504, No. 4: Recognize and
       science standard 5.5 and       express the difference between



                                    175
       social studies standard         linear and exponential growth.
       6.6)


4.3.D. Procedures

4.3.D.1 – Use graphing                 Pg. 430, No. 7: Use graphing
techniques on a number line.           techniques on a number line to
    • Absolute value                   model both absolute value and
    • Arithmetic operations            arithmetic operations.
       represented by vectors          Pg. 432, No. 12: Investigate
       (arrows) (e.g., “-3 + 6” is     inequalities and nonlinear
       “left 3, right 6”)              equations informally.

4.3.D.2 – Solve simple linear          Pg. 432, No. 10: Solve simple
equations informally, graphically,     linear equations using concrete,
and using formal algebraic             informal, and graphical methods,
methods.                               as well as appropriate paper-and-
    • Multi-step, integer              pencil techniques.
       coefficients only               Pg. 432, No. 11: Explore linear
       (although answers may           equations through the use of
       not be integers)                calculators, computers, and other
    • Using paper-and-pencil,          technology.
       calculators, graphing
       calculators, spreadsheets,
       and other technology
    • Simple literal equations
       (e.g., A = lw)

4.3.D.3 – Solve simple linear          Pg. 432, No. 12: Investigate
inequalities.                          inequalities and nonlinear
                                       equations informally.




                                     176
                                   4.3.D.4 – Create, evaluate, and
                                   simplify algebraic expressions
                                   involving variables.
                                       • Order of operations,
                                          including appropriate use
                                          of parentheses
                                       • Distributive property
                                       • Substitution of a number
                                          for a variable
                                       • Translation of a verbal
                                          phrase or sentence into an
                                          algebraic expression,
                                          equation, or inequality,
                                          and vice versa

                                   4.3.D.5 – Understand and apply        Pg. 432, No. 12: Investigate
                                   the properties of operations,         inequalities and nonlinear
                                   numbers, equations, and               equations informally.
                                   inequalities.

                                      •   Additive inverse
                                      •   Multiplicative inverse
                                      •   Addition and
                                          multiplication properties
                                          of equality
                                      •   Addition and
                                          multiplication properties
                                          of inequalities

4.4 All students will develop an  4.4.A. Data Analysis
    understanding of the concepts
    and techniques of data        4.4.A.1 – Select and use               Pg. 394, No. 9: Generate,
    analysis, probability, and    appropriate representations for        collect, organize, and analyze


                                                                       177
    discrete mathematics, and     sets of data, and measures of          data and represent this data in
    will use them to model        central tendency (mean, median,        tables, charts, and graphs.
    situations, solve problems,   and mode).                             Pg. 394, No. 10: Select and use
    and analyze and draw              • Type of display most             appropriate graphical
    appropriate inferences from           appropriate for given data     representations and measures of
    data.                             • Box-and-whisker plot,            central tendency (mean, mode,
4.4.A. Data Analysis                      upper quartile, lower          and median) for sets of data.
4.4.B. Probability                        quartile
4.4.C. Discrete Mathematics –         • Scatter plot
Systematic Listing and Counting       • Calculators and computer
4.4.D. Discrete Mathematics –             used to record and
Vertex-Edge Graphs and                    process information
Algorithms                            • Finding the median and
                                          mean (weighted average)
                                          using frequency data
                                      • Effect of additional data
                                          on measures of central
                                          tendency

                                  4.4.A.2 – Make inferences and          Pg. 395, No. 11: Make
                                  formulate and evaluate                 inferences and formulate and
                                  arguments based on displays and        evaluate arguments based on data
                                  analysis of data sets.                 analysis and data displays.

                                  4.4.A.3 – Estimate lines of best       Pg. 395, No. 12: Use lines of
                                  fit and use them to interpolate        best fit to interpolate and predict
                                  within the range of the data.          from data.

                                  4.4.A.4 – Use surveys and
                                  sampling techniques to generate
                                  data and draw conclusions about
                                  large groups.



                                                                       178
4.4.B. Probability                     Pgs. 392-393, Overview

4.4.B.1 – Interpret probabilities      Pg. 396, No. 16: Interpret
as ratios, percents, and decimals.     probabilities as ratios and
                                       percents.

4.4.B.2 – Determine probabilities      Pg. 396, No. 13: Determine the
of compound events.                    probability of a compound event.


4.4.B.3 – Explore the
probabilities of conditional
events (e.g., if there are seven
marbles in a bag, three red and
four green, what is the
probability that two marbles
picked from the bag, without
replacement, are both red).

4.4.B.4 – Model situations             Pg. 396, No. 15: Use models of
involving probability with             probability to predict events
simulations (using spinners, dice,     based on actual data.
calculators and computers) and
theoretical models.
    • Frequency, relative
        frequency

4.4.B.5 – Estimate probabilities       Pg. 396, No. 14: Model
and make predictions based on          situations involving probability,
experimental and theoretical           such as genetics, using both
probabilities.                         simulations and theoretical
                                       methods.




                                     179
4.4.B.6 – Play and analyze
probability-based games, and
discuss the concepts of fairness
and expected value.

4.4.C. Discrete Mathematics –         Pgs. 471-472, Overview
Systematic Listing and
Counting

4.4.C.1 – Apply the                   Pg. 473, No. 6: Use systematic
multiplication principle of           listing, counting, and reasoning
counting.                             in a variety of different contexts.
    • Permutations: ordered
       situations with
       replacement (e.g., number
       of possible license plates)
       vs. ordered situations
       without replacement (e.g.,
       number of possible slates

       of 3 class officers from a
       23-student class)
   •   Factorial notation
   •   Concept of combinations
       (e.g., number of possible
       delegations of 3 out of 23
       students)


4.4.C.2 – Explore counting            Pg. 476, No. 9: Explore methods
problems involving Venn               for storing, processing, and
diagrams with three attributes        communicating information.
(e.g., there are 15, 20, and 25


                                    180
students respectively in the chess
club, the debating team, and the
engineering society; how many
different students belong to the
three clubs if there are 6 students
in chess and debating, 7 students
in chess and engineering, 8
students in debating and
engineering, and 2 students in all
three?).


4.4.C.3 – Apply techniques of           Pg. 359, No. 9: Use patterns,
systematic listing, counting, and       relationships, and functions to
reasoning in a variety of different     model situations and to solve
contexts.                               problems in mathematics and in
                                        other subject areas.


4.4.D. Discrete Mathematics –
Vertex-Edge Graphs and
Algorithms

4.4.D.1 – Use vertex-edge graphs        Pg. 474, No. 7: Recognize
and algorithmic thinking to             common discrete mathematical
represent and find solutions to         models, explore their properties,
practical problems.                     and design them for specific
    • Finding the shortest              situations.
        network connecting              Pg. 477, No. 10: Devise,
        specified sites                 describe, and test algorithms for
    • Finding a minimal route           solving optimization and search
        that includes every street      problems.
        (e.g., for trash pick-up)


                                      181
•   Finding the shortest route
    on a map from one site to
    another
•   Finding the shortest
    circuit on a map that
    makes a tour of specified
    sites
•   Limitations of computers
    (e.g., the number of
    routes for a delivery truck
    visiting n sites is n!, so
    finding the shortest circuit
    by examining all circuits
    would overwhelm the
    capacity of any computer,
    now or in the future, even
    if n is less than 100)




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STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM
SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING,
REPRESENTATIONS,   AND   TECHNOLOGY     TO   SOLVE
PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

Descriptive Statement: The mathematical processes described here
highlight ways of acquiring and using the content knowledge and skills
delineated in the first four mathematics standards.

Problem Solving: Problem posing and problem solving involve examining
situations that arise in mathematics and other disciplines and in common
experiences, describing these situations mathematically, formulating
appropriate mathematical questions, and using a variety of strategies to find
solutions. Through problem solving, students experience the power and
usefulness of mathematics. Problem solving is interwoven throughout the
grades to provide a context for learning and applying mathematical ideas.

Communication: Communication of mathematical ideas involves students’
sharing their mathematical understandings in oral and written form with
their classmates, teachers, and parents. Such communication helps students
clarify and solidify their understanding of mathematics and develop
confidence in themselves as mathematics learners. It also enables teachers
to better monitor student progress.

Connections: Making connections involves seeing relationships between
different topics, and drawing on those relationships in future study. This
applies within mathematics, so that students can translate readily between
fractions and decimals, or between algebra and geometry; to other content
areas, so that students understand how mathematics is used in the sciences,
the social sciences, and the arts; and to the everyday world, so that students
can connect school mathematics to daily life.

Reasoning: Mathematical reasoning is the critical skill that enables a
student to make use of all other mathematical skills. With the development
of mathematical reasoning, students recognize that mathematics makes sense
and can be understood. They learn how to evaluate situations, select
problem-solving strategies, draw logical conclusions, develop and describe
solutions, and recognize how those solutions can be applied.



                                     183
Representations: Representations refers to the use of physical objects,
drawings, charts, graphs, and symbols to represent mathematical concepts
and problem situations. By using various representations, students will be
better able to communicate their thinking and solve problems. Using
multiple representations will enrich the problem solver with alternative
perspectives on the problem. Historically, people have developed and
successfully used manipulatives (concrete representations such as fingers,
base ten blocks, geoboards, and algebra tiles) and other representations (such
as coordinate systems) to help them understand and develop mathematics.

Technology: Calculators and computers need to be used along with other
mathematical tools by students in both instructional and assessment
activities. These tools should be used, not to replace mental math and paper-
and-pencil computational skills, but to enhance understanding of
mathematics and the power to use mathematics. Students should explore
both new and familiar concepts with calculators and computers and should
also become proficient in using technology as it is used by adults (e.g., for
assistance in solving real-world problems).




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                        Mathematics 4.5 Process Standard

At each grade level, with respect to content appropriate for that grade level, students will:

   A. Problem Solving
        1. Learn mathematics through problem solving, inquiry, and discovery.
        2. Solve problems that arise in mathematics and in other contexts (cf.
           workplace readiness standard 8.3).
                • Open-ended problems
                • Non-routine problems
                • Problems with multiple solutions
                • Problems that can be solved in several ways
        3. Select and apply a variety of appropriate problem-solving strategies (e.g.,
           “try a simpler problem” or “make a diagram”) to solve problems.
        4. Pose problems of various types and levels of difficulty.
        5. Monitor their progress and reflect on the process of their problem solving
           activity.
        6. Distinguish relevant from irrelevant information, and identify missing
           information.
   B. Communication
        1. Use communication to organize and clarify their mathematical thinking.
                • Reading and writing
                • Discussion, listening, and questioning
        2. Communicate their mathematical thinking coherently and clearly to peers,
           teachers, and others, both orally and in writing.
        3. Analyze and evaluate the mathematical thinking and strategies of others.
        4. Use the language of mathematics to express mathematical ideas precisely.
   C. Connections
        1. Recognize recurring themes across mathematical domains (e.g., patterns in
           number, algebra, and geometry).
        2. Use connections among mathematical ideas to explain concepts (e.g., two
           linear equations have a unique solution because the lines they represent
           intersect at a single point).
        3. Recognize that mathematics is used in a variety of contexts outside of
           mathematics.
        4. Apply mathematics in practical situations and in other disciplines.
        5. Trace the development of mathematical concepts over time and across
           cultures (cf. world languages and social studies standards).
        6. Understand how mathematical ideas interconnect and build on one another
           to produce a coherent whole.
                                             185
D. Reasoning
     1. Recognize that mathematical facts, procedures, and claims must be justified.
     2. Use reasoning to support their mathematical conclusions and problem
        solutions.
     3. Select and use various types of reasoning and methods of proof.
     4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the
        correctness of their problem solutions.
     5. Make and investigate mathematical conjectures.
             • Counterexamples as a means of disproving conjectures
             • Verifying conjectures using informal reasoning or proofs.
     6. Evaluate examples of mathematical reasoning and determine whether they
        are valid.
E. Representations
     1. Create and use representations to organize, record, and communicate
        mathematical ideas.
            • Concrete representations (e.g., base-ten blocks or algebra tiles)
            • Pictorial representations (e.g., diagrams, charts, or tables)
            • Symbolic representations (e.g., a formula)
            • Graphical representations (e.g., a line graph)
     2. Select, apply and translate among mathematical representations to solve
        problems.
     3. Use representations to model and interpret physical, social, and
        mathematical phenomena.
F. Technology
     1. Use technology to gather, analyze, and communicate mathematical
        information.
     2. Use computer spreadsheets, software, and graphing utilities to organize and
        display quantitative information.
     3. Use graphing calculators and computer software to investigate properties of
        functions and their graphs.
     4. Use calculators and problem-solving tools (e.g., to explore patterns, to
        validate solutions).
     5. Use computer software to make and verify conjectures about geometric
        objects.
     6. Use computer-based technology for mathematical applications in the
        sciences (cf. science standards).




                                       186
New Jersey Core Curriculum Content Standards
                     for
                Mathematics
  (Text underlined reflects changes adopted on January 9, 2008)




       (Arranged by Strand across All Grade Levels, and
  including Preschool Learning Expectations in Mathematics)




         New Jersey Department of Education
            Office of Academic Standards
                    January 2008


                               187
                      Revisions to New Jersey’s Mathematics Standards
                                as adopted on January 9, 2008


   The newly adopted changes may be found on pages 11, 17, 19, 23, 27, 31, 33, 40, and 45.
Additions have been underlined and deletions have been bracketed.
    On page 11, the word “rates” is added to 4.1.8A3; and a new bullet, “Least common multiple,
greatest common factor,” is added to clarify 4.1.6A7.
     On page 17, three new cumulative progress indicators (CPIs) are added. The new 4.2.8A7
specifies that students will “Create two-dimensional representations (e.g., nets or projective views) for
the surfaces of three-dimensional objects.” The new 4.2.8A6 and 4.2.12A5 add the content that
students will “Perform basic geometric constructions using a variety of methods (e.g., straightedge and
compass, patty/tracing paper, or technology.”
     On page 19, 4.2.12B1 and 4.2.12C1 are edited slightly for clarification. The new 4.2.12C3 adds
the content that students will be able to “Find an equation of a circle given its center and radius and,
given an equation of a circle in standard form, find its center and radius.”
    On page 23, a new bullet, “Special right triangles,” is added to 4.2.12E1.
     On page 27, a new bullet, “Solutions of systems of linear inequalities using graphing techniques,”
is added to 4.3.12B2; also the original portion of the CPI is edited to explicitly include both algebraic
and graphing techniques. The seventh bullet is shortened to “Solutions of systems of equations.”
Combined with the inclusion of both algebraic and graphing techniques, the intent is to now include
the algebraic solution of systems of equations. In the first bullet of this CPI, the slope of a “curve” is
deleted.
     On page 31, three CPIs are edited. A new bullet, “literal equations,” is added to both 4.3.8D2 and
4.3.12D2. CPI 4.3.12D2 is further edited to clarify the inclusion of inequalities. Two new bullets,
(“Perform simple operations with rational expressions” and “Evaluate polynomial and rational
expressions”) are added to 4.2.12D1.
     On page 33, 4.4.8A2 is edited to clarify the inclusion of data “sets.” A new bullet, “Correlation
vs. causation,” is added to 4.4.12A2. The new 4.4.12A6 adds the content that students will
“Distinguish between randomized experiments and observational studies.”
    On page 40, the new 4.5A6 adds the content that at each grade level, with respect to content
appropriate for that grade level, students will “Distinguish relevant from irrelevant information, and
identify missing information.”
     On page 45, the 2004 Achieve, Inc. document, “Ready or Not: Creating a High School Diploma
That Counts,” is added to the list of references. This document is available online at
http://www.achieve.org/publications/national_reports_view and includes the American Diploma
Project Benchmarks in Mathematics.
   Additional information concerning New Jersey’s participation in the American Diploma Project
may be found at
                            www.njhighschoolsummit.org/events.asp

                                         (Inside front cover)

 Adopted January 9, 2008                          188                  Adopted January 9, 2008
                                        PREFACE

     This document is a newly formatted version of the New Jersey Core Curriculum
     Content Standards for Mathematics, as revised and adopted by the New Jersey State
     Board of Education in July 2002 and revised in January 2008. It was developed in
     response to requests from schools and school districts for a version that would make
     it easier to track the learning of specific mathematics content across grade levels.

     The mathematics content and numbering of the cumulative progress indicators in this
     version of the standards remain unchanged from the version adopted by the State
     Board of Education in July 2002. Consequently, in order to align related content
     across grades, the indicators within a particular grade level have sometimes been
     arranged out of numerical order.

     The descriptive statements accompanying each of the five standards have been
     broken up into pieces, each of which now accompanies the lettered strand to which it
     refers. In all cases, however, it is the formatting and arrangement that are new; the
     content remains unchanged. It is also worth emphasizing that the goal remains
     unchanged:

            To enable ALL of New Jersey’s children to acquire the
            mathematical skills, understandings, and attitudes that they will
            need to be successful in their careers and daily lives.

     The New Jersey Core Curriculum Content Standards are intended for all students.
     This includes students who are college-bound or career-bound, gifted and talented,
     those whose native language is not English, students with disabilities, and students
     from diverse socioeconomic backgrounds. State Board adoption of the revised Core
     Curriculum Content Standards for Mathematics means that every student will be
     involved in experiences addressing all of the expectations set forth in the standards.
     It does not mean that all students will be enrolled in the same courses. Different
     groups of students should address the standards at different levels of depth and may
     complete the core curriculum according to different timetables. Depending on their
     interests, abilities, and career plans, many students will and should develop
     knowledge and skills that go beyond the specific indicators of the Core Curriculum
     Content Standards.

     Finally, the answers to a series of frequently asked questions concerning the revised
     standards are available on the Department’s website. For the convenience of those
     receiving this document, the questions and answers have been reprinted here,
     following the content of the last mathematics standard. For additional information,
     including suggested teaching strategies for implementing these standards, and for
     sample assessment items linked to the Statewide assessments, educators are
     encouraged to explore the Department’s website, at http://www.state.nj.us/education/.




Adopted January 9, 2008                       189                  Adopted January 9, 2008
           STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS)
                ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL
           OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.

           Number Sense. Number sense is an intuitive feel for numbers and a common sense approach to using them. It is a comfort with
           what numbers represent that comes from investigating their characteristics and using them in diverse situations. It involves an
           understanding of how different types of numbers, such as fractions and decimals, are related to each other, and how each can best be
           used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called numeration
           and thus includes the important concepts of place value, number base, magnitude, and approximation and estimation.
            Preschool Learning               4.1.2A. Number Sense                   4.1.3 A. Number Sense 4.1.4 A. Number Sense 4.1.5 A. Number Sense
               Expectations                               Grade 2                              Grade 3                                 Grade 4                                 Grade 5
 EXPECTATION 1:                              By the end of Grade 2,                 Building upon knowledge and           Building upon knowledge and             Building upon knowledge and
          Children                           students will:                         skills gained in preceding            skills gained in preceding              skills gained in preceding
  demonstrate an                                                                    grades, by the end of Grade 3,        grades, by the end of Grade 4,          grades, by the end of Grade 5,
  understanding of number                                                           students will:                        students will:                          students will:
  and numerical operations.
1.1 Demonstrates understanding of            1. Use real-life experiences, 1. Use real-life experiences, 1. Use real-life experiences, 1. Use real-life experiences,
one-to-one correspondence (e.g.,                physical materials, and       physical materials, and       physical materials, and       physical materials, and
places one placemat at each place,
gives each child one cookie, places             technology to construct       technology to construct       technology to construct       technology to construct
one animal in each truck, hands out             meanings for numbers          meanings for numbers          meanings for numbers          meanings for numbers
manipulatives to be shared with a               (unless otherwise noted, all          (unless otherwise noted, all           (unless otherwise noted, all            (unless otherwise noted, all
friend saying "One for you, one for me.").      indicators for grade 2 pertain to     indicators for grade 3 pertain to      indicators for grade 4 pertain to       indicators for grade 5 pertain to
                                                these sets of numbers as well).       these sets of numbers as well).        these sets of numbers as well).         these sets of numbers as well).
 1.3 Learns to say the                         •       Whole numbers                • Whole numbers through                • Whole numbers through               [Exploration of negative numbers is
 counting numbers.                                  through hundreds                    hundred thousands                     millions                           included in 4.1.4 A 7 below.]
                                               •      Ordinals                      • Commonly used fractions              • Commonly used fractions               • All fractions as part of a whole,
                                               •       Proper fractions            (denominators of 2, 3, 4, 5,               (denominators of 2, 3, 4, 5, 6, 8,      as subset of a set, as a location
                                                    (denominators of 2, 3,         6, 8, 10) as part of a whole,              10, 12, and 16) as part of a            on a number line, and as
                                                                                   as a subset of a set, and as a             whole, as a subset of a set, and        divisions of whole numbers
                                                    4, 8, 10)
                                                                                   location on a number line                  as a location on a number line
                                                                      4. Explore the extension of the place value           • Decimals through hundredths           • All decimals
                                                                         system to decimals through hundredths.
 1.5 Recognizes and names                    2. Demonstrate an under-          2. Demonstrate an under-                   2. Demonstrate an under-                [Use of concrete representations
 some written numerals.                         standing of whole number           standing of whole number                  standing of place value              (e.g., base-ten blocks) is included in
                                                place value concepts.              place value concepts.                     concepts.                            indicator 4.5 E 1.]
                                                                                    3. Identify whether any whole         3. Demonstrate a sense of the 3. Demonstrate a sense of the
                                                                                       number is odd or even.                relative magnitudes of numbers. relative magnitudes of numbers.
1.4 Discriminates numbers from                                                                                                 [Recognizing orders of magnitude associated with large and small
other symbols in the environment                                                                                                 physical quantities is included in science indicator 5.3.4 A 2.]
(e.g., street signs, license plates,
room number, clock, etc.).
                                             3. Understand that numbers             5. Understand the various             4. Understand the various
[According to Preschool Health, Safety
                                                have a variety of uses.                uses of numbers.                      uses of numbers.
and Physical Education Expectation 3.5
Knows how to dial 911 for help.]
                                             4. Count and perform simple             • Counting, measuring,                • Counting, measuring,
 1.2 Spontaneously counts for
                                                computations with coins.                labeling (e.g., numbers on             labeling (e.g., numbers on
 own purposes (e.g., counting                                                           baseball uniforms)
 blocks or cars, counting                                                                                                      baseball uniforms),
 beads while stringing them,                  • Amounts up to $1.00                   [Counting money is also                  locating (e.g., Room 235 2. Recognize the decimal nature
                                                (using cents notation)                included in indicators                   is on the second floor)     of United States currency and
 handing out napkins).                                                                4.1.3 B 5 and 4.1.4 B 6.]                                            compute with money.
                                                                                                                          5. Use concrete and pictorial           4. Use whole numbers,
                                                                                                                             models to relate whole numbers,         fractions, and decimals to
                                                                                                                             commonly used fractions, and            represent equivalent forms
                                                                                                                             decimals to each other, and to
                                                                                                                             represent equivalent forms of the       of the same number.
                                                                                                                             same number.
                                                                                                                                                                 5. Develop and apply number
                                                                                                                                                                     theory concepts in problem
                                                                                                                                                                     solving situations.
                                                                                                                                                                    • Primes, factors, multiples


 1.6 Compares numbers in                     5. Compare and order                   6. Compare and order                  6. Compare and order                    6. Compare and order
 different contexts                             whole numbers.                         numbers.                              numbers.                                numbers.
 (e.g., using words such as                                                                                               7. Explore settings that give
  more and less).                                                                                                                                                 [Use of integers is included in



                    Adopted January 9, 2008                                                       190                              Adopted January 9, 2008
                                 rise to negative numbers. science indicator 5.3.4 A 3.]
                                •   Temperatures below 0o,
                                  debts
                                • Extension of the number
                                  line




Adopted January 9, 2008   191         Adopted January 9, 2008
                                                                                           4.1 NUMBER AND NUMERICAL OPERATIONS

Descriptive Statement: Numbers and arithmetic operations are what most of the general public think about when they think
of mathematics; and, even though other areas like geometry, algebra, and data analysis have become increasingly important in
recent years, numbers and operations remain at the heart of mathematical teaching and learning. Facility with numbers, the
ability to choose the appropriate types of numbers and the appropriate operations for a given situation, and the ability to
perform those operations as well as to estimate their results, are all skills that are essential for modern day life.


4.1.6 A. Number Sense                    4.1.7 A. Number Sense                  4.1.8 A. Number Sense                     4.1.12 A. Number Sense
              Grade 6                                Grade 7                                      Grade 8                            Grade 12
Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills
gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the
end of Grade 6, students will:     end of Grade 7, students will:     end of Grade 8, students will:     end of Grade 12, students will:

1. Use real-life experiences,   1. Extend understanding of the   1. Extend understanding of the   1. Extend understanding of the
   physical materials, and         number system by                 number system by                 number system to all real
   technology to construct         constructing meanings for the    constructing meanings for the    numbers.
   meanings for numbers (unless    following (unless otherwise      following (unless otherwise
     otherwise noted, all indicators        noted, all indicators for grade 7         noted, all indicators for grade 8
     for grade 6 pertain to these sets      pertain to these sets of                  pertain to these sets of
     of numbers as well).                   numbers as well):                         numbers as well):
 •     All integers

 •     All fractions as part of a         • Rational numbers                      •   Rational numbers
     whole, as subset of a set, as a      • Percents                              •   Percents
     location on a number line, and                                               •   Exponents
     as divisions of whole numbers                                                •   Roots
                                                                                  •   Absolute values
 •     All decimals                       • Whole numbers with exponents          • Numbers represented in scientific
                                                                                       notation



3. Demonstrate a sense of the      2. Demonstrate a sense of the      2. Demonstrate a sense of the
   relative magnitudes of numbers.    relative magnitudes of numbers.    relative magnitudes of numbers.
                                 6. Understand that all fractions   6. Recognize that repeating decimals correspond to
                                    can be represented as repeating    fractions and determine their fractional equivalents.
                                    or terminating decimals.          • 5/7 = 0. 714285714285… = 0. 714285
4. Explore the use of ratios and 3. Understand and use ratios,      3. Understand and use ratios,
   proportions in a variety of       proportions, and percents         rates, proportions, and
   situations.                       (including percents greater       percents (including percents
5. Understand and use whole-         than 100 and less than 1) in a    greater than 100 and less than
   number percents between 1 and     variety of situations.            1) in a variety of situations.
     100 in a variety of situations.
2. Recognize the decimal nature
     of United States currency and
     compute with money.
6. Use whole numbers,                    5. Use whole numbers,                  5. Use whole numbers,                      [Relate to indicator 4.5 E 2,
   fractions, and decimals to               fractions, decimals, and               fractions, decimals, and                select, apply, and translate
   represent equivalent forms of            percents to represent                  percents to represent                   among           mathematical
   the same number.                         equivalent forms of the same           equivalent forms of the same            representations to solve
                                            number.                                number.                                 problems.]
7. Develop and apply number
   theory concepts in problem
   solving situations.
  • Primes, factors, multiples
  • Common multiples, common factors
  • Least common multiple, greatest common factor
8. Compare and order numbers. 4. Compare and order numbers                      4. Compare and order numbers              2. Compare and order rational
                                     of all named types.                           of all named types.                       and irrational numbers.
                                          [Use of graphing techniques           7. Construct meanings for                 3. Develop conjectures and
                                          on a number line is included             common irrational numbers,                informal proofs of properties
                                          in indicator 4.3.7 D 1.]                 such as π (pi) and the square             of number systems and sets
                                                                                   root of 2.                                of numbers.


       Adopted January 9, 2008                                              192                             Adopted January 9, 2008
                 Numerical Operations. Numerical operations are an essential part of the mathematics curriculum, especially in the
       elementary grades. Students must be able to select and apply various computational methods, including mental math, pencil-and-
       paper techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers,
       fractions, decimals, and other kinds of numbers. With the availability of calculators that perform these operations quickly and
       accurately, the instructional emphasis now is on understanding the meanings and uses of these operations, and on estimation and
       mental skills, rather than solely on the development of paper-and-pencil proficiency.
       Preschool Learning           4.1.2 B. Numerical                  4.1.3 B. Numerical               4.1.4 B. Numerical                4.1.5 B. Numerical
          Expectations              Operations       Grade 2             Operations        Grade 3       Operations        Grade 4         Operations      Grade 5
1.8 Adds two groups of              1. Develop the meanings of          1. Develop the meanings of       1. Develop the meanings of        1. Recognize the
concrete objects by counting           addition and subtraction             the four basic arithmetic       the four basic arithmetic         appropriate use of each
the total (e.g., three blue pegs,      by concretely modeling               operations by modeling          operations by modeling            arithmetic operation in
three yellow pegs, six pegs            and discussing a large               and discussing a large          and discussing a large            problem situations.
altogether).                           variety of problems.                 variety of problems.            variety of problems.
                                      • Joining, separating,              • Addition and                   • Addition and
1.9 Subtracts one group of
concrete objects from                     and comparing                       subtraction: joining,           subtraction: joining,
another by taking some                                                        separating, comparing           separating, comparing
away and then counting the          2. Explore the meanings of           • Multiplication:                • Multiplication: repeated
remainder (e.g., "I have four          multiplication and                    repeated addition,              addition, area/array
carrot sticks. I'm eating one!         division by modeling and              area/array                   • Division: repeated
Now I have 3!").                       discussing problems.              • Division: repeated                subtraction, sharing
                                                                             subtraction, sharing
                                    3. Develop proficiency with         2. Develop proficiency with      2. Develop proficiency with
                                       basic addition and subtraction       basic multiplication and        basic multiplication and
                                       number facts using a variety         division number facts           division number facts using a
                                       of fact strategies (such as          using a variety of fact         variety of fact strategies (such
                                       “counting on” and “near              strategies (such as “skip       as “skip counting” and
                                       doubles”) and then commit            counting” and “repeated         “repeated subtraction”) and
                                       them to memory.                      subtraction”).                  then commit them to memory.
[The Foundations for                4. Construct, use, and explain      3. Construct, use, and explain   3. Construct, use, and explain 2. Construct, use, and explain
performing addition and                procedures for performing           procedures for performing        procedures for performing           procedures for performing
subtraction calculations               addition and subtraction            whole number calculations        whole number calculations           addition and subtraction with
are laid through activities            calculations with:                  with:                            and with:                           fractions and decimals with:
associated with Preschool                • Pencil-and-paper                   • Pencil-and-paper              • Pencil-and-paper                  • Pencil-and-paper
Mathematics Expectations                 • Mental math                        • Mental math                   • Mental math                       • Mental math
1.8 and 1.9 above]                       • Calculator                         • Calculator                    • Calculator                        • Calculator
                                    5. Use efficient and accurate       4. Use efficient and accurate    4. Use efficient and accurate 3. Use an efficient and accurate
                                       pencil-and-paper procedures         pencil-and-paper procedures      pencil-and-paper procedures pencil-and-paper procedure
                                       for computation with whole          for computation with whole       for computation with                for division of a 3-digit
                                       numbers.                            numbers.                         whole numbers.                      number by a 2-digit number.
                                     • Addition of 2-digit numbers       • Addition of 3-digit numbers    • Addition of 3-digit numbers
                                     • Subtraction of 2-digit            • Subtraction of 3-digit         • Subtraction of 3-digit
                                         numbers                             numbers                          numbers
                                                                         • Multiplication of 2-digit      • Multiplication of 2-digit
                                                                             numbers by 1-digit               numbers
                                                                             numbers                      • Division of 3-digit numbers
                                                                                                              by 1-digit numbers
                                                                                                         5. Construct and use                [Explaining procedures for
                                                                                                              procedures for                 performing decimal addition
                                                                                                              performing decimal             and subtraction is included
                                                                                                              addition and subtraction. in 4.1.5 B 2 above.]
                                                                        5. Count and perform simple      6. Count and perform simple
                                     [Counting coins up to $1.00
                                                                           computations with money.          computations with money.
                                     (cents notation) is included
                                     in indicator 4.1.2 A 4.]             • Cents notation (¢)              • Standard dollars and
                                                                                                                 cents notation
                                    6. Select pencil-and-paper,     6. Select pencil-and-paper,          7. Select pencil-and-paper,         4. Select pencil-and-paper,
                                       mental math, or a calculator    mental math, or a calculator         mental math, or a calculator        mental math, or a calculator
                                       as the appropriate              as the appropriate                   as the appropriate                  as the appropriate
                                       computational method in a       computational method in a            computational method in a           computational method in a
                                       given situation depending       given situation depending            given situation depending           given situation depending
                                       on the context and numbers.     on the context and numbers.          on the context and numbers.         on the context and numbers.
                                                              4.1   Strand B, Numerical Operations, is continued on the next page



              Adopted January 9, 2008                                                 193                          Adopted January 9, 2008
                                                                                  4.1 NUMBER AND NUMERICAL OPERATIONS




4.1.6 B. Numerical Operations 4.1.7 B. Numerical Operations 4.1.8 B. Numerical Operations 4.1.12 B. Numerical Operations
            Grade 6                                Grade 7                            Grade 8                             Grade 12
1. Recognize the appropriate use
   of each arithmetic operation                                                  [Applying mathematics in practical situations and in
   in problem situations.                                                        other disciplines is included in indicator 4.5 C 4.]




                                                                          1. Use and explain procedures
                                                                             for performing calculations
                                                                             involving addition,
2. Construct, use, and explain        1. Use and explain procedures          subtraction, multiplication,   1. Extend understanding and use
   procedures for performing             for per-forming calculations        division, and exponentiation      of operations to real numbers
   calculations with fractions           with integers and all number        with integers and all number      and algebraic procedures.
   and decimals with:                    types named above with:             types named above with:
  • Pencil-and-paper                    • Pencil-and-paper                  • Pencil-and-paper
  • Mental math                         • Mental math                       • Mental math
  • Calculator                          • Calculator                        • Calculator
3. Use an efficient and accurate                                          .
   pencil-and-paper procedure for
   division of a 3-digit number by
   a 2-digit number.




  [Procedures for performing
  decimal multiplication and
  division are included in
  4.1.6 B 2 above.]
                                                                    [Compound interest is included in indicators
                                                             4.3.7 C 1,       4.3.8 C 2,        and        4.3.12 C 1.]


4. Select pencil-and-paper,
    mental math, or a calculator
    as the appropriate
    computational method in a
    given situation depending on
    the context and numbers.
                                     4.1   Strand B, Numerical Operations, is continued on the next page




      Adopted January 9, 2008                                           194                     Adopted January 9, 2008
                                                 4.1.2 B. Numerical Operations 4.1.3 B. Numerical Operations 4.1.4 B. Numerical Operations 4.1.5 B. Numerical Operations
                                                      Grade 2 (continued)                 Grade 3 (continued)                  Grade 4 (continued)                   Grade 5 (continued)
 No Associated Preschool Learning Expectations




                                                 7. Check the reasonableness of    7.    Check the reasonableness of 8. Check the reasonableness of    5. Check the reasonableness of
                                                    results of computations.            results of computations.         results of computations.          results of computations.
                                                                                                                     9. Use concrete models to           [Formal procedures for adding
                                                                                                                         explore addition and            and subtracting fractions are
                                                                                                                         subtraction with fractions.     included in 4.1.5 B 2 above.]
                                                 8. Understand and use the                                           10. Understand and use the        6. Understand and use the
                                                    inverse relationship between                                         inverse relationships between     various relationships among
                                                    addition and subtraction.                                            addition and subtraction and      operations and properties of
                                                                                                                                                           operations.
                                                                                                                               between multiplication
                                                                                                                         and division.




                                                           Estimation. Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily
                                                 mastered by children. It involves an educated guess about a quantity or an intelligent prediction of the outcome of a computation.
                                                 The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best
                                                 way to make that determination is through the use of strong estimation skills. Equally important is an awareness of the many
                                                 situations in which an approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these
                                                 judgments and use mathematics more powerfully as a result.
                                                 Preschool Learning     4.1.2 C.     Estimation        4.1.3 C. Estimation               4.1.4 C.    Estimation         4.1.5 C. Estimation
                                                    Expectations                   Grade 2                         Grade 3                         Grade 4                        Grade 5
                                                                        1. Judge without counting      1. Judge without counting       1. Judge without counting
                                                                           whether a set of objects        whether a set of objects        whether a set of objects
                                                                           has less than, more than,       has less than, more than,       has less than, more than,
                                                                           or the same number of           or the same number of           or the same number of
                                                                           objects as a reference set.     objects as a reference set.     objects as a reference set.
1.7 Uses estimation as a                                                3. Explore a variety of        2. Construct and use a variety  2. Construct and use a variety 1. Use a variety of
method for approximating                                                   strategies for estimating      of estimation strategies        of estimation strategies         estimation strategies for
an appropriate amount                                                      both quantities (e.g., the     (e.g., rounding and mental      (e.g., rounding and mental       both number and
(e.g., at snack time,                                                      number of marbles in a         math) for estimating both       math) for estimating both        computation.
deciding how many                                                          jar) and results of            quantities and the result of    quantities and the results of
napkins to take from a                                                     computation.                   computations.                   computations.
large pile for the group,                                                                              3. Recognize when an            3. Recognize when an             2. Recognize when an
determining number of                                                                                      estimate is appropriate,        estimate is appropriate,        estimate is appropriate,
blocks to use when                                                                                         and understand the              and understand the              and understand the
building structures).                                                                                      usefulness of an estimate       usefulness of an estimate       usefulness of an
                                                                                                           as distinct from an exact       as distinct from an exact       estimate as distinct from
                                                                                                           answer.                         answer.                         an exact answer.
                                                                        2. Determine the               4. Use estimation to            4. Use estimation to             3. Determine the
                                                                           reasonableness of an            determine whether the           determine whether the           reasonableness of an
                                                                           answer by estimating the        result of a computation         result of a computation         answer by estimating the
                                                                           result of computations          (either by calculator or        (either by calculator or        result of operations.
                                                                           (e.g., 15 + 16 is not 211).     by hand) is reasonable.         by hand) is reasonable.
                                                                                                                  [Relate to science indicator 5.3.4 A 1, determining 4. Determine whether a given
                                                                                                                  the reasonableness of estimates, measurements, and       estimate is an overestimate
                                                                                                                  computations when doing science.]                        or an underestimate.

                                                      Adopted January 9, 2008                                         195                        Adopted January 9, 2008
                                                                                     4.1 NUMBER AND NUMERICAL OPERATIONS
4.1.6 B. Numerical Operations 4.1.7 B. Numerical Operations 4.1.8 B. Numerical Operations 4.1.12 B. Numerical Operations
      Grade 6 (continued)                 Grade 7 (continued)                   Grade 8 (continued)                  Grade 12 (continued)
                                     2. Use exponentiation to find         2. Use exponentiation to find        2. Develop, apply, and explain
                                        whole number powers of                whole number powers of               methods for solving problems
                                        numbers.                              numbers.                             involving rational and
                                                                                                                   negative exponents.
5. Find squares and cubes of                                               3. Find square and cube roots of     4. Understand and apply the
   whole numbers.                                                             numbers and understand the           laws of exponents to simplify
                                                                              inverse nature of powers and         expressions involving
                                                                              roots.                               numbers raised to powers.
6. Check the reasonableness of        [Relate to Science Indicator 5.3.4 A 1, determining the reasonableness
   results of computations.           of estimates, measurements, and computations when doing science.]
                                                                           4. Solve problems involving
                                                                              proportions and percents.

7. Understand and use the
   various relationships among
   operations and properties of
   operations.

8. Understand and apply the          3. Understand and apply the           5. Understand and apply the
   standard algebraic order of          standard algebraic order of           standard algebraic order of
   operations for the four basic        operations, including                 operations, including
   operations, including                appropriate use of                    appropriate use of
   appropriate use of                   parentheses.                          parentheses.
   parentheses.
                                                                                                                3. Perform operations on
                                                                                                                   matrices.
                                                                                                                  • Addition and subtraction
                                                                                                                  • Scalar multiplication




4.1.6 C.    Estimation               4.1.7 C.    Estimation                4.1.8 C.   Estimation                4.1.12 C.    Estimation
            Grade 6                              Grade 7                               Grade 8                              Grade 12
                                                                           1. Estimate square and cube
                                                                              roots of numbers.



1. Use a variety of strategies for   1. Use equivalent                     2. Use equivalent
   estimating both quantities           representations of numbers            representations of numbers
   and the results of                   such as fractions, decimals,          such as fractions, decimals,
   computations.                        and percents to facilitate            and percents to facilitate
                                        estimation.                           estimation.

2. Recognize when an estimate                                              3. Recognize the limitations of      1. Recognize the limitations of
   is appropriate, and                                                        estimation and assess the            estimation, assess the amount
   understand the usefulness of                                               amount of error resulting            of error resulting from
   an estimate as distinct from                                               from estimation.                     estimation, and determine
   an exact answer.                                                                                                whether the error is within
                                                                                                                   acceptable tolerance limits.
3. Determine the reasonableness
   of an answer by estimating                                                  [Relate to indicator 4.5 D 4, relying on reasoning, rather than
   the result of operations.                                                   answer keys, to check the correctness of problem solutions.]


4. Determine whether a given
   estimate is an overestimate or
   an underestimate.


      Adopted January 9, 2008                                           196                           Adopted January 9, 2008
       STANDARD 4.2                (GEOMETRY AND MEASUREMENT)
             ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES,
       RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE, AND ANALYZE PHENOMENA.
                   Geometric Properties. This includes identifying, describing and classifying standard geometric objects, describing and comparing properties of geometric
       objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry,
       congruence, and similarity.
      Preschool Learning            4.2.2 A.       Geometric               4.2.3 A.      Geometric              4.2.4 A.      Geometric               4.2.5 A.      Geometric
         Expectations               Properties        Grade 2              Properties       Grade 3             Properties        Grade 4             Properties        Grade 5
EXPECTATION 2:                      By the end of Grade 2,                 Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills
                                    students will:                         gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the
Children develop knowledge                                                 end of Grade 3, students will: end of Grade 4, students will: end of Grade 5, students will:
of spatial concepts, e.g.,          1. Identify and describe spa-        1. Identify and describe         1. Identify and describe
shapes and measurement.                tial relationships among              spatial relationships of two    spatial relationships of two
                                       objects in space and their            or more objects in space.       or more objects in space.
                                       relative shapes and sizes.         • Direction, orientation,        • Direction, orientation, and
2.5 Uses positional words            • Inside/outside, left/right,            and perspectives (e.g.,          perspectives (e.g., which
in a functional way                      above/below, between                                                  object is on your left when
                                                                              which object is on your          you are standing here?)
(e.g., "I put the red block on       • Smaller/larger/same size,              left when you are
top of the cabinet.").                   wider/ narrower, longer/shorter                                   • Relative shapes and sizes
                                                                              standing here?)              • Shadows (projections) of
                                     • Congruence (i.e., same size
                                         and shape)                       • Relative shapes and sizes          everyday objects
2.1 Identifies basic shapes in      2. Use concrete objects, drawings, 2. Use properties of standard 2. Use properties of standard 2. Identify, describe, compare,
the environment (e.g., circle,         and computer graphics to             three-dimensional and           three-dimensional and two-       and classify polygons.
square, triangle, cube, sphere).       identify, classify, and describe     two-dimensional shapes          dimensional shapes to identify,  • Triangles by angles and sides
                                       standard three-dimensional                                           classify, and describe them.     • Quadrilaterals, including
                                                                            to identify, classify, and
2.6 Makes three-dimensional            and two-dimensional shapes.                                         • Vertex, edge, face, side, angle    squares, rectangles, parallelo-
                                                                            describe them.
constructions and models            • Vertex, edge, face, side                                             • 3D figures – cube, rectangular     grams, trapezoids, rhombi
                                                                          • Vertex, edge, face, side,        prism, sphere, cone, cylinder,
(e.g., sculptures that have         • 3D figures – cube, rectangular                                                                         • Polygons by number of sides.
                                                                             angle                           and pyramid
height, depth, and width).              prism, sphere, cone, cylinder,                                                                       • Equilateral, equiangular,
                                                                          • 3D figures – cube,             • 2D figures – square, rectangle,
                                       and pyramid                                                                                                        regular
                                                                              rectangular prism, sphere,           circle, triangle, quadrilateral,
2.7 Makes connections               • 2D figures – square, rectangle,                                                                                   • All points equidistant from
                                       circle, triangle                       cone, cylinder, and                  pentagon, hexagon, octagon
between two-dimensional and                                                                                                                               a given point form a circle
three-dimensional forms             • Relationships between three- and        pyramid                            • Inclusive relationships –
                                                                            • 2D figures – square,                 squares are rectangles, cubes
(e.g., circle-sphere, square-          two-dimensional shapes (i.e., the
                                                                                                                   are rectangular prisms
cube, triangle-pyramid).               face of a 3D shape is a 2D shape)      rectangle, circle, triangle,
                                                                              pentagon, hexagon,
                                                                              octagon



[Models of 3D objects are                                                                                                                             3. Identify similar figures.
included in Preschool
Mathematics Expectation
2.6 above]

[Identifying basic shapes           3. Describe, identify and              3. Identify and describe relationships 3. Identify and describe relationships 4. Understand and apply the
in the environment is                  create instances of line               among two-dimensional shapes.           among two-dimensional shapes.         concepts of congruence and
included in Preschool                  symmetry.                             • Same size, same shape                • Congruence                            symmetry (line and rotational).
                                                                             •   Lines of symmetry                • Lines of symmetry
Mathematics Expectation
2.1 above]                                                                 4. Understand and apply
                                                                                                 4. Understand and apply          1. Understand and apply concepts
                                                                              concepts involving lines,
                                                                                                    concepts involving lines,        involving lines and angles.
                                                                              angles, and circles.  angles, and circles.             •Notation for line, ray, angle,
                                                                             • Line, line segment, • Point, line, line                  line segment
                                                                               endpoint               segment, endpoint              •Properties of parallel, perpen-
                                                                                                   • Parallel, perpendicular            dicular, and intersecting lines
                                                                                                   • Angles – acute, right, obtuse •Sum of the measures of the
[Students      in     early                                                                        • Circles – diameter,                interior angles of a triangle is
elementary          grades                                                                            radius, center                    180°
sometimes confuse space-            4. Recognize, describe, extend 5. Recognize, describe,       5. Recognize, describe,
filling patterns (discussed            and create designs and patterns extend, and create space-    extend, and create space-
here) with sequential                  with geometric objects of       filling patterns.            filling patterns.
patterns discussed in                  different shapes and colors.
Preschool Mathematics
Expectations 3.5 and 3.6
and in Standard 4.3.]




              Adopted January 9, 2008                                                      197                             Adopted January 9, 2008
                                                                                                      4.2 GEOMETRY AND MEASUREMENT

Descriptive Statement: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve
describing the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling, and
measurement can help us to describe and interpret our physical environment and to solve problems.

4.2.6 A. Geometric Properties 4.2.7 A. Geometric Properties 4.2.8 A. Geometric Properties 4.2.12 A. Geometric Properties
              Grade 6                               Grade 7                                   Grade 8                                    Grade 12
Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills
gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the
end of Grade 6, students will:     end of Grade 7, students will:     end of Grade 8, students will:     end of Grade 12, students will:
6. Identify, describe, and draw the
    faces or shadows (projections) of
    three-dimensional geometric
                                                                             7. Create two-dimensional
    objects from different perspectives.                                       representations (e.g., nets
7. Identify a three-dimensional shape                                          or projective views) for the 2. Draw perspective views of
    with given projections (top, front                                         surfaces of three-              3D objects on isometric dot
    and side views).                                                           dimensional objects.            paper, given 2D
8. Identify a three-dimensional shape                                                                                          representations (e.g., nets or
    with a given net (i.e., a flat pattern                                                                                     projective views).
    that folds into a 3D shape).
2. Identify, describe, compare,         1. Understand and apply              3. Understand and apply                       1. Use geometric models to
   and classify polygons and               properties of polygons.              properties of polygons.                       represent real-world
   circles.                               • Quadrilaterals, including         • Quadrilaterals, including                     situations and objects and to
 • Triangles by angles and sides             squares, rectangles,                  squares, rectangles, parallelo-            solve problems using those
 • Quadrilaterals, including                 parallelograms, trapezoids,           grams, trapezoids, rhombi                  models (e.g., use Pythagorean
    squares, rectangles, parallelo-          rhombi                              • Regular polygons                           Theorem to decide whether
    grams, trapezoids, rhombi                                                    • Sum of measures of interior
                                          • Regular polygons                                                                  an object can fit through a
 • Polygons by number of sides                                                     angles of a polygon
                                                                                                                              doorway).
 • Equilateral, equiangular, regular                                             • Which polygons can be used
 • All points equidistant from a                                                   alone to generate a tessellation
    given point form a circle                                                      and why
5. Compare properties of cylinders, prisms,
   cones, pyramids, and spheres.
                                                                             2. Understand and apply the
                                                                                Pythagorean theorem.
3. Identify similar figures.            2. Understand and apply the          4. Understand and apply the
                                           concept of similarity.               concept of similarity.
                                          • Using proportions to find          • Using proportions to find
                                             missing measures                         missing measures
                                         • Scale drawings                        • Scale drawings
                                         • Models of 3D objects                  • Models of 3D objects
4. Understand and apply the                                                                                                3. Apply the properties of geometric shapes.
   concepts of congruence and                                                                                                • Parallel lines – transversal, alternate
   symmetry (line and rotational).                                                                                             interior angles, corresponding angles
                                                                                                                             • Triangles
1. Understand and apply concepts                                            1. Understand and apply concepts                   a. Conditions for congruence
   involving lines and angles.                                                 involving lines, angles, and planes.            b. Segment joining midpoints of
  • Notation for line, ray,                                                   • Complementary and                                 two sides is parallel to and half
      angle, line segment                                                           supplementary angles                          the length of the third side
  • Properties of parallel, perpen-                                              • Vertical angles                             c. Triangle Inequality
      dicular, and intersecting lines                                            • Bisectors and perpendicular bisectors     • Minimal conditions for a shape to
                                                                                 • Parallel, perpendicular, and                be a special quadrilateral
  • Sum of the measures of the                                                      intersecting planes                      • Circles – arcs, central and inscribed
      interior angles of a triangle                                              • Intersection of plane with cube,            angles, chords, tangents
      is 180°                                                                       cylinder, cone, and sphere               • Self-similarity
                                                                            6. Perform basic geometric               5. Perform basic geometric
                                                                             constructions using a variety of constructions using a variety of
                                                                             methods (e.g., straightedge and          methods (e.g., straightedge and compass,
                                                                             compass, patty/tracing paper, or         patty/tracing paper, or technology).
                                                                             technology).                               • Perpendicular bisector of a line segment
                                                                                 • Congruent angles or line segments    • Bisector of an angle
                                                                                 • Midpoint of a line segment           • Perpendicular or parallel lines
                                        3. Use logic and reasoning to        5. Use logic and reasoning to           4. Use reasoning and some form of proof to
                                           make and support conjectures         make and support conjectures            verify or refute conjectures and theorems.
                                           about geometric objects.             about geometric objects.              • Verification or refutation of proposed proofs
                                                                                                                      • Simple proofs involving congruent triangles
                                                                                                                      • Counterexamples to incorrect conjectures

       Adopted January 9, 2008                                             198                             Adopted January 9, 2008
                  Transforming Shapes. Analyzing how various transformations affect geometric objects allows students to enhance their spatial sense.
        This includes combining shapes to form new ones and decomposing complex shapes into simpler ones. It includes the standard geometric
        transformations of translation (slide), reflection (flip), rotation (turn), and dilation (scaling). It also includes using tessellations and fractals to create
        geometric patterns.
       Preschool Learning               4.2.2 B.   Transforming          4.2.3 B.    Transforming       4.2.4 B.    Transforming              4.2.5 B.   Transforming
          Expectations                  Shapes       Grade 2             Shapes       Grade 3           Shapes        Grade 4                 Shapes      Grade 5
[Identifying patterns is included       1. Use simple shapes to                                         1. Use simple shapes to
in Preschool Mathematics                   make designs, patterns,                                         cover an area
Expectation 3.5 below]                     and pictures.                                                   (tessellations).
                                        2. Combine and subdivide
                                           simple shapes to make
                                           other shapes.
                                                                         1. Describe and use            2. Describe and use                   1. Use a translation, a
                                                                            geometric                      geometric                             reflection, or a rotation
                                                                            transformations                transformations                       to map one figure onto
                                                                            (slide, flip, turn).           (slide, flip, turn).                  another congruent
                                                                                                                                                 figure.




3.5 Identifies patterns                                                  2. Investigate the             3. Investigate the                    2. Recognize, identify, and
in the environment                                                          occurrence of geometry         occurrence of geometry                describe geometric
(e.g., "Look at the rug.                                                    in nature and art.             in nature and art.                    relationships and
It has a circle, then a                                                                                                                          properties as they exist
number, then a                                                                                                                                   in nature, art, and other
letter...").                                                                                                                                     real-world settings.




        Coordinate Geometry. Coordinate geometry provides an important connection between geometry and algebra. It facilitates the
        visualization of algebraic relationships, as well as an analytical understanding of geometry.
        Preschool Learning        4.2.2 C. Coordinate             4.2.3 C. Coordinate             4.2.4 C. Coordinate           4.2.5 C. Coordinate
           Expectations           Geometry Grade 2                Geometry Grade 3                Geometry Grade 4              Geometry Grade 5
                                                                  1. Locate and name points 1. Locate and name points 1. Create geometric shapes
                                                                       in the first quadrant on a    in the first quadrant on a    with specified properties
                                                                       coordinate grid.              coordinate grid.              in the first quadrant on a
                                                                                                                                   coordinate grid.




 [Vocabulary to describe distances is   1. Give and follow directions                                   2. Use coordinates to give or
 included in Preschool Mathematics        for getting from one point                                       follow directions from one point
 Expectation 2.3 below]
                                          to another on a map or grid.                                     to another on a map or grid.
2.4 Uses vocabulary to
describe directional
concept (e.g., "Watch me
climb up the ladder and
slide down.").




                Adopted January 9, 2008                                                199                         Adopted January 9, 2008
                                                                                           4.2 GEOMETRY AND MEASUREMENT



4.2.6 B. Transforming Shapes 4.2.7 B. Transforming Shapes 4.2.8 B. Transforming Shapes 4.2.12 B. Transforming Shapes
            Grade 6                             Grade 7                              Grade 8                                Grade 12
                                                                       [Determining which polygons can be 3. Determine whether two or
                                                                       used alone to generate a tessellation more given shapes can be
                                                                       is included in indicator 4.2.8 A 3.]  used to generate a tessellation.
                                                [Finding the area of geometric figures made
                                                by combining other figures is included in
                                                indicators 4.2.7 E 1 and 4.2.8 E 1.]
1. Use a translation, a reflection, 2. Understand and apply             1. Understand and apply             1. Determine, describe, and draw the
   or a rotation to map one            transformations.                    transformations.                  effect of a transformation, or a sequence
   figure onto another congruent      • Finding the image, given the      • Finding the image, given the     of transformations, on a geometric or
   figure.                               pre-image, and vice-versa           pre-image, and vice-versa       algebraic [object] representation, and,
                                     • Sequence of transformations        • Sequence of transformations      conversely, determine whether and how
                                       needed to map one figure onto        needed to map one figure onto one [object]representation can be
                                       another                              another                          transformed to another by a
                                     • Reflections, rotations, and        • Reflections, rotations, and      transformation or a sequence of
                                       translations result in images        translations result in images    transformations.
                                       congruent to the pre-image           congruent to the pre-image      2. Recognize three-dimensional
                                     • Dilations (stretching/shrinking) • Dilations (stretching/shrinking) figures obtained through trans-
                                       result in images similar to the      result in images similar to the    formations of two-dimensional
                                       pre-image                            pre-image
                                                                                                               figures (e.g., cone as rotating an
                                                                                                               isosceles triangle about an
                                                                                                               altitude), using software as an aid
                                                                                                               to visualization.
2. Recognize, identify, and                                             2. Use iterative procedures to      4. Generate and analyze
   describe geometric                                                      generate geometric patterns.          iterative geometric patterns.
   relationships and properties                                           • Fractals (e.g., the Koch           • Fractals (e.g., Sierpinski’s
   as they exist in nature, art,                                            Snowflake)                             Triangle)
   and other real-world settings.                                         • Self-similarity                    • Patterns in areas and
                                                                          • Construction of initial stages         perimeters of self-similar
                                                                          • Patterns in successive stages          figures
                                                                            (e.g., number of triangles in
                                                                            each stage of Sierpinski’s         • Outcome of extending
                                                                            Triangle)                              iterative process indefinitely




4.2.6 C. Coordinate Geometry 4.2.7 C. Coordinate Geometry 4.2.8 C. Coordinate Geometry 4.2.12 C. Coordinate Geometry
             Grade 6                          Grade 7                              Grade 8                               Grade 12
1. Create geometric shapes with 1. Use coordinates in four             1. Use coordinates in four           1. Use coordinate geometry to
   specified properties in the first quadrants to represent               quadrants to represent               represent and verify properties
   quadrant on a coordinate grid.    geometric concepts.                  geometric concepts.                  of lines and line segments.
                                                                                                             •   Distance between two points
                                                                                                             •   Midpoint and slope of a line segment
                                                                                                             •   Finding the intersection of two lines
                                      [Graphing functions on the         [Developing an informal             •   Lines with the same slope are parallel
                                      coordinate plane is included       notion of slope is included         •   Lines that are perpendicular have
                                      in indicator 4.3.7 B 1.]           in indicator 4.3.8 B 1.]                 slopes whose product is –1
                                    2. Use a coordinate grid to model 2. Use a coordinate grid to model
                                       and quantify transformations      and quantify transformations
                                       (e.g., translate right 4 units).  (e.g., translate right 4 units).
                                                                                                            2. Show position and represent motion
                                                                                                               in the coordinate plane using vectors.
                                                                                                              • Addition and subtraction of vectors
                                                                                                            3. Find an equation of a circle
                                                                                                               given its center and radius
                                                                                                               and, given an equation of a
                                                                                                               circle in standard form, find
                                                                                                               its center and radius.




     Adopted January 9, 2008                                         200                        Adopted January 9, 2008
      Units of Measurement. Measurement helps describe our world using numbers. An understanding of how we attach numbers to
      real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical
      knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
      Preschool Learning      4.2.2 D. Units of                4.2.3 D. Units of             4.2.4 D. Units of                  4.2.5 D. Units of
         Expectations         Measurement Grade 2              Measurement Grade 3           Measurement Grade 4                Measurement Grade 5
3.4 Seriates objects          1. Directly compare and          1. Understand that everyday 1. Understand that everyday          [Relate to science indicator
according to various              order objects according to      objects have a variety of    objects have a variety of        5.3.4 A 1, determining the
properties including size,        measurable attributes.          attributes, each of which    attributes, each of which        reasonableness of estimates,
number, length,                • Attributes – length,             can be measured in many      can be measured in many          measurements,           and
heaviness, texture (rough           weight, capacity, time,       ways.                        ways.                            computations when doing
to smooth) or loudness.             temperature                                                                                 science.]
                              2. Recognize the need for a
                                  uniform unit of measure.
2.2 Uses standard and         3. Select and use appropriate 2. Select and use appropriate 2. Select and use appropriate         1. Select and use
nonstandard measurement           standard and non-standard       standard units of measure    standard units of measure           appropriate units to
units (e.g., measuring            units of measure and            and measurement tools to     and measurement tools to            measure angles and area.
body length with unifix           standard measurement tools      solve real-life problems.    solve real-life problems
cubes, using a tape               to solve real-life problems.
measure to gauge height         •      Length – inch, foot,      • Length – fractions of an   • Length – fractions of an
of block construction,               yard, centimeter, meter        inch (1/4, 1/2), mile,        inch (1/8, 1/4, 1/2), mile,
counting the number of                                              decimeter, kilometer          decimeter, kilometer
cups it takes to fill a                                          • Area – square inch,        • Area – square inch,
bucket with water).                                                  square centimeter            square centimeter
                                                                                              • Volume – cubic inch,
                                                                                                  cubic centimeter
                               • Weight – pound, gram,           • Weight – ounce             • Weight – ounce
                                   kilogram
                               • Capacity – pint, quart,         • Capacity – fluid ounce,    • Capacity – fluid ounce,
                                   liter                             cup, gallon, milliliter      cup, gallon, milliliter
                               • Time – second, minute,                                      5. Solve problems
                                   hour, day, week, month,                                       involving elapsed time.
                                   year
                               • Temperature – degrees
                                   Celsius, degrees Fahrenheit
                                                                                                                                2. Convert measurement
                                                                                                                                   units within a system
                                                                                                                                   (e.g., 3 feet = __ inches).
                                                                                              3. Develop and use personal 3. Know approximate
                                                                                                 referents to approximate      equivalents between the
                                                                                                 standard units of measure     standard and metric systems
                                                                                                 (e.g., a common paper         (e.g., one kilometer is
                                                                                                 clip is about an inch long).  approximately 6/10 of a mile).
[Using estimation as a        4. Estimate measures.           3. Incorporate estimation       4. Incorporate estimation in
method for approximating                                         in measurement                  measurement activities (e.g.,
an appropriate amount is                                         activities (e.g., estimate      estimate before measuring).
included in Preschool                                            before measuring).
                                                                                              [Relate to science indicator
Mathematics Expectation
                                                                                              5.3.4 B 1 Select appropriate
1.7 above]
                                                                                              measuring        instruments
                                                                                              based on the degree of
                                                                                              precision required.]
2.3 Uses vocabulary to                                                                                                          4. Use measurements and
    describe distances                                                                                                             estimates to describe
    (e.g., "It was a really                                                                                                        and compare
    long walk to the                                                                                                               phenomena.
    playground.").




             Adopted January 9, 2008                                        201                         Adopted January 9, 2008
                                                                                              4.2 GEOMETRY AND MEASUREMENT



4.2.6 D. Units of Measurement 4.2.7 D. Units of Measurement 4.2.8 D. Units of Measurement 4.2.12 D. Units of Measurement
            Grade 6                             Grade 7                                Grade 8                         Grade 12




1. Select and use appropriate
   units to measure angles, area,
   surface area, and volume.


2. Use a scale to find a distance
   on a map or a length on a
   scale drawing.




                                    1. Solve problems requiring           1. Solve problems requiring
                                       calculations that involve              calculations that involve
3. Convert measurement units           different units of measurement         different units of measurement
   within a system (e.g.,              within a measurement system            within a measurement system
   3 feet = ___ inches).               (e.g., 4’3” plus 7’10” equals          (e.g., 4’3” plus 7’10” equals
                                       12’1”).                                12’1”).
4. Know approximate equivalents                                           2. Use approximate equivalents
    between the standard and                                                  between standard and metric
    metric systems (e.g., one                                                 systems to estimate
    kilometer is approximately                                                measurements (e.g., 5
    6/10 of a mile).                                                          kilometers is about 3 miles).
                                    3. Recognize that all               5. Recognize that all             1. Understand and use the
                                       measurements of continuous          measurements of continuous         concept of significant digits.
                                       quantities are approximations.      quantities are approximations.
                                                                        3. Recognize that the degree      2. Choose appropriate tools and
                                                                           of precision needed in            techniques to achieve the
                                                                           calculations depends on how       specified degree of precision
                                                                           the results will be used and      and error needed in a
                                                                           the instruments used to           situation.
                                                                           generate the measurements.       • Degree of accuracy of a given
5. Use measurements and             2. Select and use appropriate units 4. Select and use appropriate          measurement tool
   estimates to describe and           and tools to measure quantities     units and tools to measure       • Finding the interval in which a
   compare phenomena.                  to the degree of precision          quantities to the degree of         computed measure (e.g., area
                                       needed in a particular problem-     precision needed in a               or volume) lies, given the
                                       solving situation.                  particular problem-solving          degree of precision of linear
                                                                           situation.                          measurements
                                                                        6. Solve problems that involve
                                                                            compound measurement
                                                                            units, such as speed (miles
                                                                            per hour), air pressure
                                                                            (pounds per square inch), and
                                                                            population density (persons
                                                                            per square mile).

      Adopted January 9, 2008                                           202                        Adopted January 9, 2008
                Measuring Geometric Objects. This area focuses on applying the knowledge and understandings of units of
       measurement in order to actually perform measurement. While students will eventually apply formulas, it is important that they
       develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students
       apply indirect measurement skills, using, for example, similar triangles and trigonometry.

       Preschool Learning       4.2.2 E.     Measuring      4.2.3 E. Measuring             4.2.4 E.    Measuring           4.2.5 E.  Measuring
          Expectations              Geometric Objects           Geometric Objects              Geometric Objects               Geometric Objects
                                          Grade 2                    Grade 3                         Grade 4                       Grade 5
[Use of nonstandard measure-   2. Directly measure the area 1. Determine the area of       1. Determine the area of
ment units is included in         of simple two-dimensional    simple two-dimensional         simple two-dimensional
Preschool       Mathematics       shapes by covering them      shapes on a square grid.       shapes on a square grid.
Expectation 2.2 above]            with squares.
                                                                                                                           1. Use a protractor to
                                                                                           [Relate to Science Indicator       measure angles.
                                                                                           5.3.4 B 2 Use a variety of
                                                                                           measuring instruments and
                                                                                           record measured quantities
                                                                                           using the appropriate units.]


                                1. Directly measure the      2. Determine the perimeter    2. Distinguish between          2. Develop and apply
                                   perimeter of simple          of simple shapes by           perimeter and area and          strategies and formulas
                                   two-dimensional shapes.      measuring all of the          use each appropriately          for finding perimeter
                                                                sides.                        in problem-solving              and area.
                                                                                              situations.                    • Square
                                                                                                                             • Rectangle



                                                                                                                           3. Recognize that
                                                                                                                              rectangles with the
                                                                                                                              same perimeter do
                                                                                                                              not necessarily have
                                                                                                                              the same area and
                                                                                                                              vice versa.
[Comparing numbers in                                        3. Measure and compare        3. Measure and compare
context (e.g., using words                                      the volume of                 the volume of
such as more and less) is                                       three-dimensional             three-dimensional
included    in    Preschool                                     objects using materials       objects using materials
Mathematics Expectation 1.6                                     such as rice or cubes.        such as rice or cubes.
above]




                                                                                                                           4. Develop informal ways
                                                                                                                              of approximating the
                                                                                                                              measures of familiar
                                                                                                                              objects (e.g., use a grid
                                                                                                                              to approximate the area of
                                                                                                                              the bottom of one’s foot).




             Adopted January 9, 2008                                     203                      Adopted January 9, 2008
                                                                                              4.2 GEOMETRY AND MEASUREMENT




4.2.6 E.                              4.2.7 E.                            4.2.8 E.                               4.2.12 E.
Measuring Geometric Objects           Measuring Geometric Objects         Measuring Geometric Objects            Measuring Geometric Objects
          Grade 6                               Grade 7                             Grade 8                                Grade 12
                       [Finding area is included in indicators
                       4.2.6 E 2 and 4.2.7 E 1 below.]

1. Use a protractor to                                                                                           1. Use techniques of indirect
   measure angles.                                                                                                  measurement to represent and
                                                                                                                    solve problems.
                                                                                                                   • Similar triangles
                                                                                                                   • Pythagorean theorem
                                                                                                                   • Right triangle trigonometry
                                                                                                                      (sine, cosine, tangent)
                                                                                                                   • Special right triangles
2. Develop and apply strategies   1. Develop and apply strategies    1. Develop and apply strategies             2. Use a variety of strategies to
   and formulas for finding          for finding perimeter and area.    for finding perimeter and                   determine perimeter and area
   perimeter and area.              • Geometric figures made by         area.                                       of plane figures and surface
  • Triangle, square, rectangle,       combining triangles,            • Geometric figures made by                  area and volume of 3D figures.
     parallelogram, and trapezoid      rectangles and circles or          combining triangles,                     • Approximation of area using
  • Circumference and area of a        parts of circles                   rectangles and circles or                  grids of different sizes
     circle                         • Estimation of area using            parts of circles                        • Finding which shape has
                                       grids of various sizes          • Estimation of area using                   minimal (or maximal) area,
                                                                          grids of various sizes                    perimeter, volume, or surface
4. Recognize that shapes with                                                                                       area under given conditions
                                                                       • Impact of a dilation on the
   the same perimeter do not                                                                                        using graphing calculators,
                                                                          perimeter and area of a
   necessarily have the same                                                                                        dynamic geometric software,
                                                                          2-dimensional figure
   area and vice versa.                                                                                             and/or spreadsheets
                                                                                                                  • Estimation of area, perimeter,
                                                                                                                    volume, and surface area
                                      2. Recognize that the volume of     2. Recognize that the volume of
                                                                                                                  [Relate to indicator 4.2.12 B 2,
                                         a pyramid or cone is one-third      a pyramid or cone is one-third       recognizing three-dimensional
                                         of the volume of the prism or       of the volume of the prism or        figures obtained through trans-
                                         cylinder with the same base         cylinder with the same base          formations of two-dimensional
                                         and height (e.g., use rice to       and height (e.g., use rice to        figures (e.g., cone as rotating an
                                         compare volumes of figures          compare volumes of figures           isosceles triangle about an
                                         with same base and height).         with same base and height).          altitude)]
3. Develop and apply strategies                                           3. Develop and apply strategies
   and formulas for finding the                                              and formulas for finding the
   surface area and volume of                                                surface area and volume of a
   rectangular prisms and                                                    three-dimensional figure.
   cylinders.                                                               • Volume - prism, cone, pyramid          [Finding surface area and
                                                                            • Surface area - prism (triangular       volume of 3D figures is
                                                                               or rectangular base), pyramid         included in indicator
                                                                               (triangular or rectangular            4.2.12 E 2 above.]
                                                                               base)
                                                                            • Impact of a dilation on the
                                                                               surface area and volume of a
                                                                               three-dimensional figure
                                                                          4. Use formulas to find the
                                                                             volume and surface area of
                                                                             a sphere.
5. Develop informal ways of
   approximating the measures
   of familiar objects (e.g., use a
   grid to approximate the area
   of the bottom of one’s foot).

Students of all ages should realize that geometry and measurement are all around them. Through study of these areas and their
applications, they should come to better understand and appreciate the role of mathematics in their lives.

      Adopted January 9, 2008                                         204                          Adopted January 9, 2008
      STANDARD 4.3               (PATTERNS AND ALGEBRA)
            ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES
      AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.

      Patterns. Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students
      should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to make mathematical
      discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to
      understand and represent mathematical and other real-world phenomena.
      Preschool Learning          4.3.2 A. Patterns           4.3.3 A. Patterns               4.3.4 A. Patterns              4.3.5 A. Patterns
         Expectations                      Grade 2                     Grade 3                          Grade 4                       Grade 5
EXPECTATION 3:                    By the end of Grade 2,      Building upon knowledge         Building upon knowledge        Building upon knowledge
 Children understand              students will:              and skills gained in            and skills gained in           and skills gained in
 patterns, relationships                                      preceding grades, by the end    preceding grades, by the end   preceding grades, by the end
                                                              of Grade 3, students will:      of Grade 4, students will:     of Grade 5, students will:
 and classification.
3.5 Identifies patterns in        1. Recognize, describe,     1. Recognize, describe,         1. Recognize, describe,        1. Recognize, describe,
the environment                      extend, and create          extend, and create              extend, and create             extend, and create
(e.g., "Look at the rug. It          patterns.                   patterns.                       patterns.                      patterns involving
has a circle, then a number,                                                                                                    whole numbers.
then a letter...").
3.6 Represents patterns in         • Using concrete
a variety of ways                    materials
(e.g., stringing beads               (manipulatives),
red/green/red/green/red/green,       pictures, rhythms, &
arranging buttons                    whole numbers
big/bigger/biggest, or
singing songs that follow a
simple pattern).
                                   • Descriptions using        • Descriptions using            • Descriptions using              Descriptions using
                                     words and symbols            words and number               words, number                   tables, verbal rules,
                                     (e.g., “add two” or “+       sentences/expressions          sentences/expressions,          simple equations, and
                                     2”)                                                         graphs, tables, variables       graphs
                                                                                                 (e.g., shape, blank, or
                                                                                                 letter)
                                   • Repeating patterns                                        • Sequences that stop or
                                                                                                 that continue infinitely
                                   • Whole number                 Whole number patterns        • Whole number patterns
                                     patterns that grow or        that grow or shrink as         that grow or shrink as
                                     shrink as a result of        a result of repeatedly         a result of repeatedly
                                     repeatedly adding or         adding, subtracting,           adding, subtracting,
                                     subtracting a fixed          multiplying by, or             multiplying by, or
                                     number (e.g., skip           dividing by a fixed            dividing by a fixed
                                     counting forward or          number                         number
                                     backward)                    (e.g., 5, 8, 11, . . . or      (e.g., 5, 8, 11, . . . or
                                                                  800, 400, 200, . . .)          800, 400, 200, . . .)
                                                                                                • Sequences can often
                                                                [Use of calculators to
                                                                                                   be extended in more
                                                                explore patterns is
                                                                included in indicator              than one way (e.g.,
                                                                4.5 F 4.]                          the next term after 1,
                                                                                                   2, 4, . . . could be 8,
                                                                                                   or 7, or … )




             Adopted January 9, 2008                                        205                         Adopted January 9, 2008
                                                                                                        4.3 PATTERNS AND ALGEBRA

Descriptive Statement: Algebra is a symbolic language used to express mathematical relationships. Students need to
understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those
relationships. Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as
solving equations, with a more visual perspective, with graphs of equations displayed on a screen. Students can then focus on
understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation.


4.3.6 A. Patterns                   4.3.7 A. Patterns                    4.3.8 A. Patterns                     4.3.12 A. Patterns
            Grade 6                             Grade 7                                 Grade 8                           Grade 12
Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills
gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the
end of Grade 6, students will:     end of Grade 7, students will:     end of Grade 8, students will:     end of Grade 12, students will:

1. Recognize, describe, extend,     1. Recognize, describe, extend,      1. Recognize, describe, extend,       1. Use models and algebraic
   and create patterns involving       and create patterns involving        and create patterns involving         formulas to represent and
   whole numbers and rational          whole numbers, rational              whole numbers, rational               analyze sequences and series.
   numbers.                            numbers, and integers.               numbers, and integers.




 • Descriptions using tables,        • Descriptions using tables,            • Descriptions using tables,      • Explicit formulas for nth
   verbal rules, simple                verbal and symbolic rules,              verbal and symbolic rules,        terms
   equations, and graphs               graphs, simple equations or             graphs, simple equations or
                                       expressions                             expressions


 •   Formal iterative formulas       • Finite and infinite sequences         • Finite and infinite sequences
   (e.g., NEXT = NOW * 3)
 • Recursive patterns,                                                       • Arithmetic sequences             • Sums of finite arithmetic
   including Pascal’s Triangle                                                 (i.e., sequences generated by      series
   (where each entry is the sum                                                repeated addition of a fixed
   of the entries above it) and                                                number, positive or negative)
   the Fibonacci Sequence:                                                   • Geometric sequences              • Sums of finite and infinite
   1, 1, 2, 3, 5, 8, . . . (where                                              (i.e., sequences generated by      geometric series
   NEXT = NOW + PREVIOUS)                                                      repeated multiplication by a
                                                                               fixed positive ratio, greater
                                                                               than 1 or less than 1)
                                    • Generating sequences by                • Generating sequences by
                                        using calculators to                   using calculators to
                                        repeatedly apply a formula             repeatedly apply a formula



                                                                                                               2. Develop an informal notion of
                                                                                                                  limit.
                                                                                                               3. Use inductive reasoning to
                                                                                                                  form generalizations.




     Adopted January 9, 2008                                           206                        Adopted January 9, 2008
Functions and Relationships. The function concept is one of the most fundamental unifying ideas of modern mathematics.
Students begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to
abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they
have observed. In high school, they use the more formal language of algebra to describe these relationships.

                                                4.3.2 B.                        4.3.3 B.                          4.3.4 B.                         4.3.5 B.
                                                Functions and Relationships Functions and Relationships Functions and Relationships                Functions and Relationships
                                                           Grade 2                          Grade 3                          Grade 4                          Grade 5




                                                1. Use concrete and pictorial   1. Use concrete and pictorial     1. Use concrete and pictorial    2. Graph points satisfying a
                                                   models of function              models to explore the basic       models to explore the basic      function from T-charts,
                                                   machines to explore the         concept of a function.            concept of a function.           from verbal rules, and from
                                                   basic concept of a function.                                                                       simple equations.
                                                                                  • Input/output tables, T-charts   • Input/output tables, T-
                                                                                                                       charts
No Associated Preschool Learning Expectations




                                                                                                                    • Combining two function       1. Describe arithmetic
                                                                                                                       machines                       operations as functions,
                                                                                                                    • Reversing a function            including combining
                                                                                                                       machine                        operations and reversing
                                                                                                                                                      them.




                                                                                      [Transformations                                              [Translations          and
                                                                                      are introduced in                                             reflections are introduced
                                                                                      indicator 4.2.3 B 1                                           in    indicator   4.2.5 B 1
                                                                                      above]                                                        above]




                                                Adopted January 9, 2008                                     207                        Adopted January 9, 2008
                                                                                             4.3 PATTERNS AND ALGEBRA




4.3.6 B.                      4.3.7 B.                       4.3.8 B.                        4.3.12 B.
Functions and Relationships Functions and Relationships Functions and Relationships Functions and Relationships
           Grade 6                       Grade 7                        Grade 8                              Grade 12
                                                                                             1. Understand relations and functions
                                                                                                and select, convert flexibly among,
                                                                                                and use various representations for
                                                                                                them, including equations or
                                                                                                inequalities, tables, and graphs.
1. Describe the general        1. Graph functions, and       1. Graph functions, and         2. Analyze and explain the general
   behavior of functions given    understand and describe       understand and describe         properties and behavior of functions
   by formulas or verbal rules    their general behavior.       their general behavior.         [of one variable] or relations, using
   (e.g., graph to determine                                                                    [appropriate] algebraic and
   whether increasing or         • Equations involving two    • Equations involving two         graphing [technologies] techniques.
   decreasing, linear or not).      variables                   variables
                                                              • Rates of change (informal     • Slope of a line [or curve]
                                                                notion of slope)
                                                                                              • Domain and range
                                                                                              • Intercepts
                                                                                              • Continuity
                                                                                              • Maximum/minimum
                                                                                              • Estimating roots of equations
                                                                                              • [Intersecting points as] Solutions
                                                                                                of systems of equations
                                                                                              • Solutions of systems of linear
                                                                                                 inequalities using graphing
                                                                                                 techniques
                                                                                               • Rates of change
                                                                                             3. Understand and perform
                                                                                                transformations on commonly-used
                                                                                                functions.
                                                                                               • Translations, reflections, dilations
                                                                                               • Effects on linear and quadratic
                                                                                                  graphs of parameter changes in
                                                                                                  equations
                                                                                               • Using graphing calculators or
                                                                                                  computers for more complex
                                                                                                  functions
                                                             2. Recognize and describe the   4. Understand and compare the
                                                                difference between linear       properties of classes of functions,
                                                                and exponential growth,         including exponential, polynomial,
                                                                using tables, graphs, and       rational, and trigonometric
                                                                equations.                      functions.
                                                                                               • Linear vs. non-linear
                                                                                               • Symmetry
                                                                                               • Increasing/decreasing on an
                                                                                                  interval




     Adopted January 9, 2008                                  208                      Adopted January 9, 2008
       Modeling. Algebra is used to model real situations and answer questions about them. This use of algebra requires the ability to
       represent data in tables, pictures, graphs, equations or inequalities, and rules. Modeling ranges from writing simple number
       sentences to help solve story problems in the primary grades to using functions to describe the relationship between two variables,
       such as the height of a pitched ball over time. Modeling also includes some of the conceptual building blocks of calculus, such as
       how quantities change over time and what happens in the long run (limits).
       Preschool Learning       4.3.2 C.   Modeling           4.3.3 C.   Modeling           4.3.4 C.      Modeling           4.3.5 C.     Modeling
          Expectations                  Grade 2                       Grade 3                           Grade 4                         Grade 5
4.2 Describes the sequence of   1. Recognize and describe     1. Recognize and describe     1. Recognize and describe        2. Draw freehand sketches
the daily routine and              changes over time (e.g.,      change in quantities.          change in quantities.           of graphs that model real
demonstrates understanding         temperature, height).        • Graphs representing         • Graphs representing             phenomena and use such
of basic temporal relations                                       change over time (e.g.,        change over time (e.g.,        graphs to predict and
(e.g., "We will go outside                                        temperature, height)           temperature, height)           interpret events.
after snack time.").                                                                          • How change in one              • Changes over time
                                                                                                 physical quantity can         • Rates of change (e.g.,
[Understanding that living
things change as they grow                                                                       produce a corresponding          when is plant growing
is included in Preschool                                                                         change in another (e.g.,         slowly/rapidly, when is
Science Expectation 3.3]                                                                         pitch of a sound depends         temperature dropping
                                                                                                 on the rate of vibration)        most rapidly/slowly)
                                2. Construct and solve        2. Construct and solve        2. Construct and solve           1. Use number sentences to
                                   simple open sentences         simple open sentences          simple open sentences           model situations.
                                   involving addition or         involving addition or          involving any one              • Using variables to
                                   subtraction.                  subtraction                    operation                         represent unknown
                                  • Result unknown               (e.g., 3 + 6 = __,             (e.g., 3 x 6 = __,                quantities
                                     (e.g., 6 – 2 = __ or        n = 15 – 3,                    n = 15 ÷ 3,                    • Using concrete materials,
                                     n = 3 + 5)                  3 + __ = 3,                    3 x __ = 0,                       tables, graphs, verbal
                                  • Part unknown                 16 – c = 7).                   16 – c = 7).                      rules, algebraic
                                     (e.g., 3 + = 8)                                                                              expressions/equations




             Adopted January 9, 2008                                     209                       Adopted January 9, 2008
                                                                                                        4.3 PATTERNS AND ALGEBRA




4.3.6 C.       Modeling              4.3.7 C. Modeling                   4.3.8 C. Modeling                  4.3.12 C. Modeling
               Grade 6                             Grade 7                            Grade 8                             Grade 12
2.   Draw freehand sketches of       1. Analyze functional               1. Analyze functional              2. Analyze and describe how a
     graphs that model real pheno-       relationships to explain how a     relationships to explain how a      change in an independent
     mena and use such graphs to         change in one quantity can         change in one quantity can          variable leads to change in a
     predict and interpret events.       result in a change in another,     result in a change in another,      dependent one.
 •     Changes over time                 using pictures, graphs, charts,    using pictures, graphs, charts,
 •     Relations between quantities      and equations.                     and equations.
 •     Rates of change (e.g., when                                                                          3. Convert recursive formulas to
       is plant growing                                                                                         linear or exponential
       slowly/rapidly, when is                                                                                  functions (e.g., Tower of
       temperature dropping most                                                                                Hanoi and doubling).
       rapidly/slowly)
1.   Use patterns, relations, and    2. Use patterns, relations,         2. Use patterns, relations,        1. Use functions to model real-
     linear functions to model           symbolic algebra, and linear       symbolic algebra, and linear        world phenomena and solve
     situations.                         functions to model situations.     functions to model situations.      problems that involve varying
 •     Using variables to represent    • Using manipulatives, tables,      • Using concrete materials           quantities.
       unknown quantities                  graphs, verbal rules,              (manipulatives), tables,        • Linear, quadratic, exponential,
 •     Using concrete materials,           algebraic expressions/             graphs, verbal rules,              periodic (sine and cosine), and
       tables, graphs, verbal rules,       equations/inequalities             algebraic expressions/             step functions (e.g., price of
       algebraic expressions/                                                 equations/inequalities             mailing a first-class letter over
       equations/inequalities          • Growth situations, such as        • Growth situations, such as          the past 200 years)
                                           population growth and              population growth and           • Direct and inverse variation
                                           compound interest, using           compound interest, using        • Absolute value
                                           recursive (e.g., NOW-              recursive (e.g., NOW-           • Expressions, equations and
                                           NEXT) formulas (cf.                NEXT) formulas (cf. science        inequalities
                                           science standards and social       standards and social studies    • Same function can model
                                           studies standards)                 standards)                         variety of phenomena
                                                                                                              • Growth/decay and change in
                                                                                                                    the natural world
                                                                                                                 • Applications in mathematics,
                                                                                                                    biology, and economics
                                                                                                                    (including compound interest)




       Adopted January 9, 2008                                         210                        Adopted January 9, 2008
        Procedures. Techniques for manipulating algebraic expressions – procedures – remain important, especially for students who may
        continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying
        properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and
        inequalities, and solving equations and inequalities.

        Preschool Learning        4.3.2 D.   Procedures         4.3.3 D.  Procedures         4.3.4 D.    Procedures         4.3.5 D.   Procedures
           Expectations                     Grade 2                      Grade 3                        Grade 4                       Grade 5

                                                                  [Use of a number line        [Use of a number line          [Use of a number line
                                                                  to construct meanings        to construct meanings          to construct meanings
                                                                  for numbers at this          for numbers at this            for numbers at this
                                                                  grade level is included      grade level is included        grade level is included
                                                                  in indicator 4.1.3 A 1.]     in indicators 4.1.4 A 1        in indicator 4.1.5 A 1.]
                                                                                               and 4.1.4 A 7.]

                                                                                                                            1. Solve simple linear
                                                                                                                               equations with
                                                                                                                               manipulatives and
                                                                                                                               informally
                                                                                                                              • Whole-number
                                                                                                                                 coefficients only,
                                                                                                                                 answers also whole
                                                                                                                                 numbers
                                                                                                                              • Variables on one side of
                                                                                                                                 equation


[Comparing numbers in                                           2. Understand and use the    2. Understand and use the
different contexts (e.g., using                                    concepts of equals, less     concepts of equals, less
words such as more and less)                                       than, and greater than to    than, and greater than in
is included in Preschool                                           describe relations           simple number
Mathematics Expectation 1.6                                        between numbers.             sentences.
above]                                                            • Symbols ( = , < , > )      • Symbols ( = , < , > )
                                  1. Understand and apply       1. Understand and apply      1. Understand, name, and
                                      (but don’t name) the         the properties of            apply the properties of
                                      following properties of      operations and numbers.      operations and numbers.
                                      addition:
                                    •    Commutative             • Commutative                 • Commutative
                                       (e.g., 5 + 3 = 3 + 5)       (e.g., 3 x 7 = 7 x 3)         (e.g., 3 x 7 = 7 x 3)
                                   • Zero as the identity        • Identity element for        • Identity element for
                                     element                       multiplication is 1           multiplication is 1
                                     (e.g., 7 + 0 = 7)             (e.g., 1 x 8 = 8)             (e.g., 1 x 8 = 8)
                                   • Associative (e.g.,                                        • Associative (e.g.,
                                     7 + 3 + 2 can be found                                      2 x 4 x 25 can be
                                     by first adding either                                      found by first
                                     7 + 3 or 3 + 2)                                             multiplying either
                                                                                                 2 x 4 or 4 x 25)
                                                                                               • Division by zero is
                                                                                                 undefined
                                                                 • Any number multiplied       • Any number
                                                                   by zero is zero               multiplied by zero is
                                                                                                 zero




              Adopted January 9, 2008                                      211                       Adopted January 9, 2008
                                                                                                        4.3 PATTERNS AND ALGEBRA




4.3.6 D.   Procedures              4.3.7 D.    Procedures                4.3.8 D.      Procedures          4.3.12 D. Procedures
           Grade 6                              Grade 7                                 Grade 8                          Grade 12
                                   1. Use graphing techniques on a       1. Use graphing techniques on a
  [Use of a number line to            number line.                           number line.
  construct meanings for             • Absolute value                      • Absolute value
  numbers at this grade level        • Arithmetic operations               • Arithmetic operations
  is included in indicator             represented by vectors                 represented by vectors
  4.1.6 A 1.]                          (arrows)                               (arrows)
                                       (e.g., “-3 + 6” is “left 3,            (e.g., “-3 + 6” is “left 3,
                                       right 6”)                              right 6”)
1. Solve simple linear equations   2. Solve simple linear equations      2. Solve simple linear equations 2. Select and use appropriate
   with manipulatives and             informally and graphically.            informally, graphically, and     methods to solve equations and
   informally.                       • Multi-step, integer                   using formal algebraic           inequalities.
  • Whole-number coefficients           coefficients only (although          methods.                        • Linear equations and
     only, answers also whole           answers may not be               • Multi-step, integer                  inequalities – algebraically
     numbers                            integers)                             coefficients only (although    • Quadratic equations – factoring
  • Variables on one or both         • Using paper-and-pencil,                answers may not be integers)      (including trinomials when
     sides of equation                  calculators, graphing              • Simple literal equations           the coefficient of x2 is 1) and
                                        calculators, spreadsheets,             (e.g., A = lw)                   using the quadratic formula
                                        and other technology                                                 • Literal equations
                                                                           • Using paper-and-pencil,
                                                                                                             • All types of equations and
                                                                               calculators, graphing
                                                                               calculators, spreadsheets,
                                                                                                                inequalities using graphing,
                                                                                                                computer, and graphing
                                                                               and other technology
                                                                                                                calculator techniques
4. Extend understanding and use                                          3. Solve simple linear
  of inequality.                                                             inequalities.                  [Use of concrete representations
 • Symbols ( ≥ , ≠ , ≤ )                                                                                    (e.g., algebra tiles) is included in
                                                                                                            indicator 4.5 E 1.]


2. Understand and apply the      4. Understand and apply the             5. Understand and apply the
    properties of operations and    properties of operations,               properties of operations,
    numbers.                        numbers, equations, and                 numbers, equations, and
  • Distributive property           inequalities.                           inequalities.
  • The product of a number • Additive inverse                             • Additive inverse
     and its reciprocal is 1       • Multiplicative inverse                • Multiplicative inverse
                                                                           • Addition and multiplication
                                                                              properties of equality
                                                                           • Addition and multiplication
                                                                              properties of inequalities
3. Evaluate numerical              3. Create, evaluate, and simplify     4. Create, evaluate, and simplify   1. Evaluate and simplify
   expressions.                       algebraic expressions                 algebraic expressions               expressions.
                                      involving variables.                  involving variables.               • Add and subtract
                                     • Order of operations, including      • Order of operations, including      polynomials
 [The distributive property             appropriate use of parentheses        appropriate use of parentheses   • Multiply a polynomial by a
 appears in 4.3.6 D 2 above.]       • Substitution of a number for a        • Distributive property              monomial or binomial
                                        variable                            • Substitution of a number for a   • Divide a polynomial by a
                                                                              variable                           monomial
                                                                            • Translation of a verbal phrase   • Perform simple operations
                                                                               or sentence into an algebraic       with rational expressions
                                                                               expression, equation, or         • Evaluate polynomial and
                                                                               inequality, and vice versa
                                                                                                                   rational expressions
                                                                                                               3. Judge the meaning, utility, and
                                                                                                                  reasonableness of the results
                                                                                                                  of symbol manipulations,
                                                                                                                  including those carried out by
                                                                                                                  technology.

Algebra is a gatekeeper for the future study of mathematics, science, the social sciences, business, and a host of other areas. In
the past, algebra has served as a filter, screening people out of these opportunities. For New Jersey to be part of the global
society, it is important that algebra play a major role in a mathematics program that opens the gates for all students.

     Adopted January 9, 2008                                          212                         Adopted January 9, 2008
         STANDARD 4.4                  (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS)
               ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF
         DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL
         SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA.

         Data Analysis (or Statistics). In today’s information-based world, students need to be able to read, understand, and interpret data
         in order to make informed decisions. In the early grades, students should be involved in collecting and organizing data, and in
         presenting it using tables, charts, and graphs. As they progress, they should gather data using sampling, and should increasingly be
         expected to analyze and make inferences from data, as well as to analyze data and inferences made by others.
         Preschool Learning             4.4.2 A.    Data Analysis        4.4.3 A.     Data Analysis          4.4.4 A.    Data Analysis        4.4.5 A.    Data Analysis
            Expectations                           Grade 2                           Grade 3                            Grade 4                          Grade 5
EXPECTATION 4:                           By the end of Grade 2,          Building upon knowledge and         Building upon knowledge and      Building upon knowledge and
 Children develop                        students will:                  skills gained in preceding          skills gained in preceding       skills gained in preceding
 knowledge of sequence                                                   grades, by the end of Grade 3,      grades, by the end of Grade 4,   grades, by the end of Grade 5,
 and temporal awareness.                                                 students will:                      students will:                   students will:
                                        1. Collect, generate, record, 1. Collect, generate,                  1. Collect, generate,         1. Collect, generate,
[Classifying objects by sorting             and organize data in                organize, and display data      organize, and display data    organize, and display
them into subgroups by one or               response to questions,              in response to questions,       in response to questions,     data.
more attributes is included in              claims, or curiosity.               claims, or curiosity.           claims, or curiosity.
Preschool         Mathematics
Expectation 3.2 below]                    • Data collected from students’ • Data collected from the            • Data collected from the     • Data generated from
                                              everyday experiences                classroom environment           school environment           surveys
                                          • Data generated from
                                              chance devices, such as
                                              spinners and dice
4.3 Arranges pictures of events in      2. Read, interpret, construct, 2. Read, interpret, construct,        2. Read, interpret, construct, 2. Read, interpret, select,
temporal order (e.g., first, a photo       and analyze displays of              analyze, generate               analyze, generate               construct, analyze,
of the child eating breakfast;             data.                                questions about, and draw       questions about, and draw       generate questions about,
second, a photo of the child                                                    inferences from displays        inferences from displays        and draw inferences from
getting on the bus; third, a photo                                              of data.                        of data.                        displays of data.
of the child in the classroom).           • Pictures, tally chart, pictograph, • Pictograph, bar graph,        • Pictograph, bar graph, • Bar graph, line graph,
                                            bar graph, Venn diagram               table                           line plot, line graph, table    circle graph, table
[Seriating objects according              • Smallest to largest,                                               • Average (mean),               • Range,
to     various     properties                 most frequent (mode)                                                most frequent (mode),           median, and
including size, number,                                                                                           middle term (median)            mean
length, heaviness, texture
                                                                         [Interpreting information in
(rough to smooth) or loudness
                                                                         graphs,      charts,      and
is included in Preschool
                                                                         diagrams is included in
Mathematics Expectation 3.4                                              language      arts   literacy
below]                                                                   indicator 3.1.3 G 3]




                                                                                                                                              3. Respond to questions
                                                                                                                                                 about data and generate
                                                                                                                                                 their own questions and
                                                                                                                                                 hypotheses.




                Adopted January 9, 2008                                               213                           Adopted January 9, 2008
                                                 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS

Descriptive Statement: Data analysis, probability, and discrete mathematics are important interrelated areas of applied
mathematics. Each provides students with powerful mathematical perspectives on everyday phenomena and with important
examples of how mathematics is used in the modern world. Two important areas of discrete mathematics are addressed in this
standard; a third area, iteration and recursion, is addressed in Standard 4.3 (Patterns and Algebra).




4.4.6 A.    Data Analysis           4.4.7 A.    Data Analysis             4.4.8 A.   Data Analysis            4.4.12 A.     Data Analysis
            Grade 6                             Grade 7                              Grade 8                               Grade 12
Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills
gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the
end of Grade 6, students will:     end of Grade 7, students will:     end of Grade 8, students will:     end of Grade 12, students will:

1. Collect, generate, organize,                                                                               1. Use surveys and sampling
   and display data.                                                                                             techniques to generate data
                                                                                                                 and draw conclusions about
                                                                                                                 large groups.
 • Data generated from surveys                                                                                  • Advantages/disadvantages of
                                                                                                                   sample selection methods
                                                                                                                   (e.g., convenience sampling,
                                                                                                                   responses to survey, random
                                                                                                                   sampling)
2. Read, interpret, select,         1. Select and use appropriate         1. Select and use appropriate       2. Evaluate the use of data in
    construct, analyze, generate        representations for sets of          representations for sets of         real-world contexts.
    questions about, and draw           data, and measures of central        data, and measures of central
    inferences from displays of         tendency (mean, median, and          tendency (mean, median, and
    data.                               mode).                               mode).
  • Bar graph, line graph, circle     • Type of display most                • Type of display most              • Accuracy and reasonableness
     graph, table, histogram             appropriate for given data            appropriate for given data         of conclusions drawn
  • Range, median, and mean           • Box-and-whisker plot, upper         • Box-and-whisker plot, upper       • Correlation vs. causation
                                         quartile, lower quartile              quartile, lower quartile
                                      • Scatter plot                        • Scatter plot
 • Calculators and computers          • Calculators and computer            • Calculators and computer          • Bias in conclusions drawn
   used to record and process            used to record and process            used to record and process          (e.g., influence of how data
   information                           information                           information                         is displayed)
                                                                            • Finding the median and            • Statistical claims based on
                                                                               mean (weighted average)             sampling
                                                                               using frequency data
                                                                            • Effect of additional data on
                                                                               measures of central tendency
                                                                          3. Estimate lines of best fit and   4. Estimate or determine lines of
                                                                             use them to interpolate within       best fit (or curves of best fit if
                                                                             the range of the data.               appropriate) with technology,
                                                                                                                  and use them to interpolate
                                                                                                                  within the range of the data.
3. Respond to questions about    2. Make inferences and                   2. Make inferences and              5. Analyze data using technology,
   data, generate their own         formulate and evaluate                   formulate and evaluate              and use statistical terminology
   questions and hypotheses, and    arguments based on displays              arguments based on displays         to describe conclusions.
   formulate strategies for         and analysis of data.                    and analysis of data sets.        • Measures of dispersion: variance,
   answering their questions and                                                                                 standard deviation, outliers
   testing their hypotheses.                                              4. Use surveys and sampling         • Correlation coefficient
                                                                             techniques to generate data      • Normal distribution (e.g., approx-
  [Interpreting and using graphic sources of information such as             and draw conclusions about          imately 95% of the sample lies
  maps, graphs, timelines, or tables to address research questions           large groups.                       between two standard deviations
  is included in language arts literacy indicator 3.1.6 H 4]                                                     on either side of the mean)
                                                                                                              3. Design a statistical
                                                                                                                 experiment, conduct the
                                                                                                                 experiment, and interpret and
                                                                                                                 communicate the outcome.
                                                                                                              6. Distinguish between
                                                                                                                 randomized experiments
                                                                                                                 and observational studies.


      Adopted January 9, 2008                                           214                      Adopted January 9, 2008
Probability. Students need to understand the fundamental concepts of probability so that they can interpret weather forecasts, avoid
unfair games of chance, and make informed decisions about medical treatments whose success rate is provided in terms of
percentages. They should regularly be engaged in predicting and determining probabilities, often based on experiments (like flipping
a coin 100 times), but eventually based on theoretical discussions of probability that make use of systematic counting strategies. High
school students should use probability models and solve problems involving compound events and sampling.

                                                4.4.2 B.   Probability            4.4.3 B.     Probability           4.4.4 B. Probability                    4.4.5 B.    Probability
                                                            Grade 2                             Grade 3                            Grade 4                                Grade 5
                                                1. Use chance devices like          1. Use everyday events and       1. Use everyday events and              3. Model situations involving
                                                   spinners and dice to explore        chance devices, such as dice,    chance devices, such as dice,           probability using simulations
                                                   concepts of probability.            coins, and unevenly divided      coins, and unevenly divided             (with spinners, dice) and
                                                                                       spinners, to explore concepts    spinners, to explore concepts           theoretical models.
                                                                                       of probability.                  of probability.
                                                  • Certain, impossible               • Likely, unlikely, certain, • Likely, unlikely, certain,
                                                                                         impossible                       impossible, improbable,
                                                                                                                          fair, unfair
                                                  • More likely, less likely, • More likely, less likely, • More likely, less likely,
                                                     equally likely                      equally likely                   equally likely
No Associated Preschool Learning Expectations




                                                                                                                       • Probability of tossing
                                                                                                                          “heads” does not depend on
                                                                                                                          outcomes of previous tosses
                                                2. Provide probability of specific                                   2. Determine probabilities of           1. Determine probabilities of
                                                   outcomes.                                                            simple events based on                  events.
                                                  • Probability of getting                                              equally likely outcomes and            • Event, probability of an event
                                                     specific outcome when coin                                         express them as fractions.             • Probability of certain event is
                                                     is tossed, when die is rolled,                                                                               1 and of impossible event is 0
                                                     when spinner is spun (e.g.,
                                                     if spinner has five equal
                                                     sectors, then probability of
                                                     getting a particular sector is
                                                     one out of five)
                                                 • When picking a marble from 2. Predict probabilities in a              3. Predict probabilities in a       2. Determine probability using
                                                   a bag with three red marbles  variety of situations (e.g.,               variety of situations (e.g.,        intuitive, experimental, and
                                                   and four blue marbles, the    given the number of items of               given the number of items of        theoretical methods (e.g.,
                                                   probability of getting a red  each color in a bag, what is               each color in a bag, what is        using model of picking items
                                                   marble is three out of seven  the probability that an item               the probability that an item        of different colors from a
                                                                                 picked will have a particular              picked will have a particular       bag).
                                                                                 color).                                    color).                            • Given numbers of various
                                                                                • What students think will                 • What students think will           types of items in a bag, what
                                                                                        happen (intuitive)                  happen (intuitive)                  is the probability that an item
                                                                                    •   Collect data and use that data    • Collect data and use that data      of one type will be picked
                                                                                        to predict the probability          to predict the probability        • Given data obtained
                                                                                        (experimental)                      (experimental)                      experimentally, what is the
                                                                                                                          • Analyze all possible outcomes       likely distribution of items in
                                                                                                                            to find the probability             the bag
                                                                                                                            (theoretical)




                                                Adopted January 9, 2008                                        215                        Adopted January 9, 2008
                                                    4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS




4.4.6 B.    Probability                4.4.7 B.   Probability               4.4.8 B.   Probability              4.4.12 B.    Probability
             Grade 6                               Grade 7                              Grade 8                             Grade 12
4. Model situations involving          2. Model situations involving        4. Model situations involving       3. Model situations involving
   probability using simulations          probability with simulations         probability with simulations        probability with simulations
   (with spinners, dice) and              (using spinners, dice,               (using spinners, dice,              (using spinners, dice,
   theoretical models                     calculators and computers)           calculators and computers)          calculators and computers)
                                          and theoretical models.              and theoretical models.             and theoretical models, and
                                                                                                                   solve problems using these
                                                                                                                   models.

                                        •   Frequency,                          • Frequency,
                                            relative frequency                    relative frequency



1. Determine probabilities of          1. Interpret probabilities as        1. Interpret probabilities as       6. Understand and use the “law
   events.                                ratios, percents, and decimals.       ratios, percents, and decimals.    of large numbers” (that
 • Event, complementary event,                                                                                     experimental results tend to
    probability of an event                                                                                        approach theoretical
 • Multiplication rule for                                                                                         probabilities after a large
    probabilities                                                                                                  number of trials).
 • Probability of certain event is 1
    and of impossible event is 0
 • Probabilities of event and
    complementary event add up
    to 1
2. Determine probability using         3. Estimate probabilities and        5. Estimate probabilities and       5. Estimate probabilities and
   intuitive, experimental, and           make predictions based on            make predictions based on           make predictions based on
   theoretical methods (e.g.,             experimental and theoretical         experimental and theoretical        experimental and theoretical
   using model of picking items           probabilities.                       probabilities.                      probabilities.
   of different colors from a
   bag).
 • Given numbers of various
   types of items in a bag, what is
   the probability that an item of
   one type will be picked
 • Given data obtained
   experimentally, what is the
   likely distribution of items in
   the bag
3. Explore compound events.                                                 2. Determine probabilities of
                                                                               compound events.
                                                                            3. Explore the probabilities of     4. Determine probabilities in
                                                                               conditional events (e.g., if        complex situations.
                                                                               there are seven marbles in a       • Conditional events
                                                                               bag, three red and four green,     • Complementary events
                                                                               what is the probability that       • Dependent and independent
                                                                               two marbles picked from the          events
                                                                               bag, without replacement, are
                                                                               both red).
5. Recognize and understand the        4. Play and analyze probability-     6. Play and analyze probability-    1. Calculate the expected value
   connections among the                  based games, and discuss the         based games, and discuss the        of a probability-based game,
   concepts of independent                concepts of fairness and             concepts of fairness and            given the probabilities and
   outcomes, picking at random,           expected value.                      expected value.                     payoffs of the various
   and fairness.                                                                                                   outcomes, and determine
                                                                                                                   whether the game is fair.
                                                                                                                2. Use concepts and formulas of
                                                                                                                   area to calculate geometric
                                                                                                                   probabilities.

      Adopted January 9, 2008                                             216                          Adopted January 9, 2008
         Discrete Mathematics—Systematic Listing and Counting. Development of strategies for listing and counting can progress
         through all grade levels, with middle and high school students using the strategies to solve problems in probability. Primary students,
         for example, might find all outfits that can be worn using two coats and three hats; middle school students might systematically list
         and count the number of routes from one site on a map to another; and high school students might determine the number of three-
         person delegations that can be selected from their class to visit the mayor.
          Preschool Learning          4.4.2C. Discrete Mathematics- 4.4.3C. Discrete Mathematics- 4.4.4C. Discrete Mathematics- 4.4.5C. Discrete Mathematics-
             Expectations             Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting
                                                Grade 2                          Grade 3                         Grade 4                            Grade 5
3.1 Sorts objects into groups
(e.g., separate basket of                  1. Sort and classify objects 1. Represent and classify data 1. Represent and classify data    [Classifying data cam be related
collected items into piles of                 according to attributes.       according to attributes,      according to attributes,      to classifying organisms, as in
pinecones, acorns and twigs).                                                such as shape or color, and   such as shape or color, and   science indicator 5.5.4 B 1, or
 [Classifying objects is included                                            relationships.                relationships.                food groups, as in Preschool
 in Expectation 3.2 below]                                                                                                               Health, Safety and Physical
3.3 Describes an object by charac-                                                                                                       Education Expectation 1.1]
                                             • Venn diagrams                • Venn diagrams               • Venn diagrams
teristics it does or does not possess
(e.g., "This button doesn't have holes.").
3.4 Seriates objects according to                                           • Numerical and               • Numerical and
various properties including size,                                             alphabetical order            alphabetical order
number, length, heaviness, texture
(rough to smooth) or loudness.
  [Counting is included in                 2. Generate all possibilities 2. Represent all possibilities 2. Represent all possibilities   1. Solve counting problems
  Preschool Mathematics                       in simple counting             for a simple counting         for a simple counting            and justify that all
  Expectations 1.3 through 1.6                situations (e.g., all outfits  situation in an organized     situation in an organized        possibilities have been
  and 1.8 above]                              involving two shirts and       way and draw conclusions      way and draw conclusions         enumerated without
                                         three pants).                  from this representation.        from this representation.          duplication.
                                                                       • Organized lists, charts        • Organized lists, charts,         • Organized lists, charts,
                                                                                                           tree diagrams                      tree diagrams, tables
3.2 Classifies objects by sorting                                                                       • Dividing into categories
them into subgroups by one or                                                                              (e.g., to find the total
more attributes (e.g., sorting                                                                             number of rectangles in
counting bears by color into trays,                                                                        a grid, find the number
separating a mixture of beans by                                                                           of rectangles of each
individual size and shape).                                                                                size and add the results)
                                                                                                                                         2. Explore the multiplication
                                                                                                                                            principle of counting in
                                                                                                                                            simple situations by
                                                                                                                                            representing all
                                                                                                                                            possibilities in an
                                                                                                                                            organized way (e.g., you
                                                                                                                                            can make 3 x 4 = 12
                                                                                                                                            outfits using 3 shirts and
                                                                                                                                            4 skirts).




                 Adopted January 9, 2008                                          217                         Adopted January 9, 2008
                                                    4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS




4.4.6 C.  Discrete Mathematics- 4.4.7 C. Discrete Mathematics- 4.4.8 C. Discrete Mathematics- 4.4.12C. Discrete Mathematics-
Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting
             Grade 6                               Grade 7                                Grade 8                               Grade 12




                                   [Venn diagrams are included in indicators
                           4.4.6 C 1,     4.4.7 C 2 ,     and      4.4.8 C 2 below.]




1. Solve counting problems and       2. Explore counting problems             2. Explore counting problems
   justify that all possibilities       involving Venn diagrams with             involving Venn diagrams with
   have been enumerated                 three attributes (e.g., there are 15,    three attributes (e.g., there are 15,
   without duplication.                 20, and 25 students respectively         20, and 25 students respectively
                                        in the chess club, the debating          in the chess club, the debating
  • Organized lists, charts, tree       team, and the engineering                team, and the engineering
     diagrams, tables                   society; how many different              society; how many different
                                        students belong to the three clubs       students belong to the three clubs
  • Venn diagrams
                                        if there are 6 students in chess         if there are 6 students in chess
                                        and debating, 7 students in chess        and debating, 7 students in chess
   [Venn        diargrams      are
                                        and engineering, 8 students in           and engineering, 8 students in
   introduced in 4.4.2 C 1.]            debating and engineering, and 2          debating and engineering, and 2
                                        students in all three?).                 students in all three?).
2. Apply the multiplication          1. Apply the multiplication              1. Apply the multiplication              2. Apply the multiplication rule
    principle of counting.               principle of counting.                   principle of counting.                  of counting in complex
  • Simple situations (e.g., you       • Permutations: ordered                  • Permutations: ordered                   situations, recognize the
     can make 3 x 4 = 12 outfits           situations with replacement              situations with replacement           difference between situations
     using 3 shirts and 4 skirts).         (e.g., number of possible                (e.g., number of possible             with replacement and without
  • Number of ways a specified             license plates) vs. ordered              license plates) vs. ordered           replacement, and recognize
     number of items can be                                                                                               the difference between
                                           situations without                       situations without
     arranged in order (concept of
     permutation)                          replacement (e.g., number                replacement (e.g., number             ordered and unordered
  • Number of ways of selecting a          of possible slates of 3 class            of possible slates of 3 class         counting situations.
     slate of officers from a class        officers from a 23 student               officers from a 23 student
     (e.g., if there are 23 students       class)                                   class)
     and 3 officers, the number is                                              • Factorial notation
     23 x 22 x 21)
3. List the possible combinations                                              • Concept of combinations           1. Calculate combinations with
   of two elements chosen from                                                   (e.g., number of possible            replacement (e.g., the number
   a given set (e.g., forming a                                                  delegations of 3 out of 23           of possible ways of tossing a
   committee of two from a                                                       students)                            coin 5 times and getting 3
   group of 12 students, finding                                                                                      heads) and without
   how many handshakes there                                                                                          replacement (e.g., number of
   will be among ten people if                                                                                        possible delegations of 3 out
   everyone shakes each other                                                                                         of 23 students).
   person’s hand once).
                                      3. Apply techniques of                3. Apply techniques of                 3. Justify solutions to counting
                                         systematic listing, counting,         systematic listing, counting,          problems.
                                         and reasoning in a variety of         and reasoning in a variety of
                                         different contexts.                   different contexts.
  [Recognizing,     describing,                                                                                    4. Recognize and explain
                                                                                                                      relationships involving
  and extending recursive
                                                                                                                      combinations and Pascal’s
  patterns, including Pascal’s                                                                                        Triangle, and apply those
  Triangle, is included in                                                                                            methods to situations
  indicator 4.3.6 A 1.]                                                                                               involving probability.


      Adopted January 9, 2008                                            218                          Adopted January 9, 2008
          Discrete Mathematics—Vertex-Edge Graphs and Algorithms. Vertex-edge graphs, consisting of dots (vertices) and lines
          joining them (edges), can be used to represent and solve problems based on real-world situations. Students should learn to follow and
          devise lists of instructions, called “algorithms,” and use algorithmic thinking to find the best solution to problems like those involving
          vertex-edge graphs, but also to solve other problems.

           Preschool Learning              4.4.2D.   Discrete Mathematics-   4.4.3D.   Discrete Mathematics-   4.4.4D.   Discrete Mathematics-   4.4.5D.   Discrete Mathematics-
              Expectations                 Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms
                                                      Grade 2                       Grade 3                               Grade 4                            Grade 5
4.1 Starts and stops on a signal           1. Follow simple sets of      1. Follow, devise, and                1. Follow, devise, and
(e.g., freezing in position                   directions (e.g., from one    describe practical sets of            describe practical sets of
when the music stops).                        location to another, or       directions (e.g., to add              directions (e.g., to add
[Following oral directions that involve       from a recipe).               two 2-digit numbers).                 two 2-digit numbers).
several actions is included in Preschool
Language Arts Literacy Expectation 1.1]
                                           3. Play simple two-person                                        2. Play two-person games      1. Devise strategies for winning
                                              games (e.g., tic-tac-toe)                                        and devise strategies for     simple games (e.g., start with
                                              and informally explore                                           winning the games (e.g.,      two piles of objects, each of
                                              the idea of what the                                             “make 5" where players        two players in turn removes
                                              outcome should be.                                               alternately add 1 or 2 and    any number of objects from a
                                                                                                               the person who reaches 5,     single pile, and the person to
                                           [According to N.J.S.A. 18A:35-4.16, “Each board of education        or another designated         take the last group of objects
                                           may offer instruction in chess during the second grade for          number, is the winner).       wins) and express those
                                           pupils in gifted and talented and special education programs.” *                                  strategies as sets of directions.
                                           4. Explore concrete models 2. Explore vertex-edge                3. Explore vertex-edge
                                              of vertex-edge graphs (e.g.        graphs.                       graphs and tree diagrams.
                                              vertices as “islands” and        • Vertex, edge                 • Vertex, edge,
                                              edges as “bridges”).                                              neighboring/adjacent,
                                                                                                                number of neighbors
                                             • Paths from one vertex to • Path                                • Path, circuit (i.e., path
                                                another                                                         that ends at its starting
                                                                                                                point)




                                           2. Color simple maps with a 3. Find the smallest number 4. Find the smallest number
                                              small number of colors.     of colors needed to color   of colors needed to color
                                                                          a map.                      a map or a graph.




                                             * [N.J.S.A. 18A:35-4.15a declares that:
                                               ”chess increases strategic thinking skills, stimulates
                                                intellectual creativity, and improves problem-solving
                                                ability while raising self esteem.”]

                 Adopted January 9, 2008                                                  219                          Adopted January 9, 2008
                                                  4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS




4.4.6D. Discrete Mathematics- 4.4.7D. Discrete Mathematics- 4.4.8D. Discrete Mathematics- 4.4.12D. Discrete Mathematics-
Vertex-Edge Graphs and Algorithms    Vertex-Edge Graphs and Algorithms     Vertex-Edge Graphs and Algorithms     Vertex-Edge Graphs and Algorithms
             Grade 6                              Grade 7                               Grade 8                              Grade 12




1. Devise strategies for winning
   simple games (e.g., start with
   two piles of objects, each of
   two players in turn removes
   any number of objects from a
   single pile, and the person to
   take the last group of objects
   wins) and express those
   strategies as sets of directions.
2. Analyze vertex-edge graphs
   and tree diagrams.
  • Can a picture or a vertex-edge
    graph be drawn with a single
    line? (degree of vertex)
  • Can you get from any vertex
    to any other vertex?
    (connectedness)
3. Use vertex-edge graphs to find 1. Use vertex-edge graphs to         1. Use vertex-edge graphs and             1. Use vertex-edge graphs and
   solutions to practical              represent and find solutions to     algorithmic thinking to                  algorithmic thinking to
   problems.                           practical problems.                 represent and find solutions to          represent and solve practical
                                                                           practical problems.                      problems.
  • Delivery route that stops at • Finding the shortest network          • Finding the shortest network            • Circuits that include every
     specified sites but involves       connecting specified sites          connecting specified sites                edge in a graph
     least travel                                                        • Finding a minimal route that            • Circuits that include every
                                                                            includes every street (e.g., for
                                                                            trash pick-up)                            vertex in a graph
                                                                         • Finding the shortest route on a         • Scheduling problems (e.g.,
  • Shortest route from one site • Finding the shortest route on a          map from one site to another              when project meetings
     on a map to another                map from one site to another     • Finding the shortest circuit on            should be scheduled to
                                     • Finding the shortest circuit on      a map that makes a tour of                avoid conflicts) using graph
                                        a map that makes a tour of          specified sites                           coloring
                                        specified sites                  • Limitations of computers                • Applications to science
                                                                             (e.g., the number of routes for a        (e.g., who-eats-whom
                                                                             delivery truck visiting n sites is       graphs, genetic trees,
                                                                             n!, so finding the shortest circuit
                                                                             by examining all circuits would          molecular structures)
                                                                               overwhelm the capacity of any
                                                                               computer, now or in the future,
                                                                               even if n is less than 100)
                                                                                                                 2. Explore strategies for making
                                                                                                                     fair decisions.
                                                                                                                  • Combining individual preferences
                                                                                                                   into a group decision (e.g.,
                                                                                                                   determining winner of an election
                                                                                                                   or selection process)
                                                                                                                 • Determining how many Student
                                                                                                                   Council representatives each
                                                                                                                   class (9th, 10th, 11th, and 12th
                                                                                                                   grade) gets when the classes have
                                                                                                                   unequal sizes (apportionment)

These topics provide students with insight into how mathematics is used by decision-makers in our society, and with important
tools for modeling a variety of real-world situations. Students will better understand and interpret the vast amounts of
quantitative data that they are exposed to daily, and they will be able to judge the validity of data-supported arguments.


      Adopted January 9, 2008                                            220                        Adopted January 9, 2008
                               Math Websites

Let’s Get Close
   - Students practice improper fractions and mixed numbers by rolling dice
http://www.math.montana.edu/mathed/distance/capstone/kershner/activities/get_cl
ose.html

Multiplication Machine
   - Students can practice multiplying from easy to megahard
http://www.amblesideprimary.com/ambleweb/mentalmaths/testtest.html

Sum Sense Division
   - Students have to complete division sentences is an allotted amount of time
http://www.oswego.org/ocsd-web/games/SumSense/sumdiv.html

Sum sense Multiplication
   - Students have to complete division sentences is an allotted amount of time
http://www.oswego.org/ocsd-web/games/SumSense/summulti.html

A great website with lots of Number Sense Interactive Games
http://edweb.tusd.k12.az.us/ekowalcz/math/elementary_web_sites.htm




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